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Appendices

Glossary

Acute angle: An angle whose measure is less than $90^\circ$. (92)

Acute triangle: A triangle all of whose angles are acute. (158)

Altitude of a prism: (1) A line segment that connects the planes of the bases of a prism and that is perpendicular to both of them. (2) The length of that line segment. (640)

Altitude of a pyramid: (1) The perpendicular line segment from the apex of a pyramid to the plane of its base. (2) The length of that line segment. (648)

Altitude of a quadrilateral that has parallel sides: (1) A perpendicular line segment that connects points on the lines of the parallel sides. (2) The length of that line segment. (359)

Altitude of a triangle: (1) A perpendicular line segment from a vertex of a triangle to the line of the opposite side. (2) The length of that line segment. (352)

Angle: A pair of rays that have the same endpoint. The angle in the figure at the right can be named ∠O or ∠AOB or ∠BOA; this notation also stands for the measure of this angle. The measure might either be found by using a protractor or worked out by the geometry of a given problem. (13)

Angle of inclination: The angle that a line forms with the horizontal. (461)

Antipodal points: The two points of intersection of a sphere with a line through its center. (691)

Apothem of a regular polygon: A perpendicular line segment from the center of the polygon to one of its sides. (574)

Area: Measure of extent for plane figures. Determined by using the Area Postulates for polygonal regions or as limits of the areas of polygons (as described for the area of the circle) for curved regions. The methods can be extended to finding the areas of surfaces in three dimensions, including the area of a sphere. (20, 664–665)

Area of a circle: The limit of the areas of inscribed regular polygons. Archimedes extended this method to find the areas of figures bounded by curves by using areas of inscribed and circumscribed polygons. (598)

blue area < curved region area < red area

Betweenness of Points: An intuitive notion, formally defined as follows by using the Ruler Postulate. A point is between two other points on the same line iff its coordinate is between their coordinates. (More briefly, A-B-C iff $a < b < c$ or $a > b > c$.) (85)

p. 729

Betweenness of Rays: An intuitive notion, formally defined as follows by using the Protractor Postulate. A ray is between two others in the same half-rotation iff its coordinate is between their coordinates. (More briefly, OA-OB-OC iff $a < b < c$ or $a > b > c$.) (93)

Birectangular quadrilateral: A quadrilateral that has two sides perpendicular to a third side. (697)

Bisector of an angle: A line or ray that divides an angle into two equal angles. (99)

Center of a regular polygon: The center of the circumscribed circle of the polygon. See Apothem of a regular polygon. (574)

Central angle of a circle: An angle whose vertex is the center of a circle. (498)

Central angle of a regular polygon: An angle formed by radii drawn to two consecutive vertices of the polygon. See Apothem of a regular polygon. (574)

Centroid of a triangle: The point in which the medians of a triangle are concurrent. Physically, a uniform triangular region balances perfectly in any position when supported at its centroid. (549)

Cevian of a triangle: A line segment that connects a vertex of a triangle to a point on the opposite side. (554)

Chord of a circle: A line segment that connects two points of a circle. (485)

Circle: The set of all points in a plane that are at a given distance from a given point in the plane. (484)

Circumcenter of a polygon: The center of the circle circumscribed about the polygon. (531)

Circumference of a circle: The limit of the perimeters of the inscribed regular polygons. (592)

Circumscribed circle about a polygon: The circle that contains all of the vertices of the polygon. (531)

Circumscribed polygon about a circle: A polygon each of whose sides is tangent to the circle. (542)

Collinear points: Points that are contained by the same line. (9)

Complementary angles: Two angles whose sum is $90^\circ$. (105)

Concave polygon: A polygon that is not convex. (258)

Concentric circles: Circles that lie in the same plane and have the same center. (484)

Concurrent lines: Lines that contain the same point. (9)

Cone: Suppose that $A$ is a plane, $R$ is a circular region in plane $A$, and $P$ is a point not in plane $A$. The solid made up of all segments that connect $P$ to a point of region $R$ is a cone. (654)

p. 730

Congruent figures: Informally, two figures are congruent if they have the same size and shape so that, if one is superimposed on the other, they match exactly. Formally, two figures are congruent if there is an isometry such that one figure is the image of the other. This definition of congruence for figures in general agrees with the definition of congruent triangles and extends the idea of same size and same shape to all plane and solid shapes. (140, 312)

Congruent triangles: Two triangles for which there is a correspondence between their vertices such that all of their corresponding sides and angles are equal. (140)

Converse of a conditional statement: The statement formed by interchanging the hypothesis and conclusion of the conditional statement. (47)

Convex polygon: A polygon such that every line segment that connects two points inside it lies entirely inside the polygon. More generally, a plane figure or a solid shape is convex iff every line segment connecting two points inside the figure or shape also lies inside it. (258)

Coplanar points: Points that are contained in the same plane. (9)

Corollary: A theorem that can be easily proved as a consequence of a definition, a postulate or another theorem. (99)

Cosine of an acute angle of a right triangle: The ratio of the length of the adjacent leg to the length of the hypotenuse. More generally, see Trigonometric functions. (455)

Cross section of a geometric solid: The intersection of a plane and the solid. (641)

Cube: A rectangular solid whose length, width, and height are equal. (629)

Cyclic polygon: A polygon for which there exists a circle that contains all of the vertices of the polygon. (530)

Cylinder: Suppose that $A$ and $B$ are two parallel planes, $R$ is a circular region in one plane, and l is a line that intersects both planes but not $R$. The solid made up of all segments parallel to line l that connect a point of region $R$ to a point of the other plane is a cylinder. (655)

Decagon: A polygon that has 10 sides. (573)

Degree measure of an arc: The measure of the central angle of an arc. (498)

Diagonal of a polygon: A line segment that connects any two nonconsecutive vertices of a polygon. (259)

Diameter of a circle: (1) A chord that contains the center of a circle. (2) The length of a chord that contains the center. (485)

Dilation: Informally, a transformation in which a figure is reduced or enlarged. (300)

p. 731

Distance between two parallel lines: In general, the length of any perpendicular segment connecting one line to the other.

Distance between two points: An intuitive idea of what we measure by using a ruler. Formalized by the Ruler Postulate. The distance between the endpoints of a line segment gives the length of the segment. (85)

Distance from a point to a line: In general, the length of the perpendicular segment from the point to the line.

Dodecagon: A polygon that has 12 sides. (585)

Dodecahedron: A polyhedron that has 12 faces. The faces of a regular dodecahedron are regular pentagons. (678)

Equiangular polygon: A convex polygon all of whose angles are equal. (572)

Equiangular triangle: A triangle all of whose angles are equal. (158)

Equilateral polygon: A polygon all of whose sides are equal. (572)

Exterior angle of a triangle: An angle that forms a linear pair with an angle of the triangle. (191)

Geometric mean: The number $b$ is the geometric mean between the numbers $a$ and $c$ iff $a$, $b$, and $c$ are positive and $\dfrac{a}{b}=\dfrac{b}{c}$. (380)

Glide reflection: A transformation that is the composite of a translation and a reflection in a line parallel to the direction of the translation. (313)

Great circle of a sphere: A set of points that is the intersection of the sphere and a plane containing its center. (690)

Heptagon: A polygon that has seven sides. (573)

Hexagon: A polygon that has six sides. (19)

Hypotenuse of a right triangle: The side opposite the right angle of the triangle. (66)

p. 732

Icosahedron: A polyhedron that has 20 faces. The faces of a regular icosahedron are equilateral triangles. (678)

Incenter of a polygon: The center of the inscribed circle of the polygon. (542)

Inscribed angle of a circle: An angle whose vertex is on a circle and whose sides each intersect the circle in another point. (505)

Inscribed circle in a polygon: A circle for which each side of the polygon is tangent to the circle. (542)

Inscribed polygon in a circle: A polygon each of whose vertices lies on the circle. (531)

Isometry: A transformation that preserves distance and angle measure. (300)

Isosceles trapezoid: A trapezoid whose legs are equal. (281)

Isosceles triangle: A triangle that has at least two equal sides. (157)

Lateral area of a prism: The sum of the areas of the lateral faces of a prism. Lateral area excludes the areas of the bases of a prism. (635)

Length of a line segment: The distance between its endpoints. (85)

Line segment: Part of a line bounded by two endpoints. If the endpoints are A and B, then the segment is written as AB (or BA). (9)

Linear pair: Two angles that have a common side and whose other sides are opposite rays. (111)

Median of a triangle: A line segment that connects a vertex of a triangle to the midpoint of the opposite side. (549)

Midpoint of a line segment: The point that divides a line segment into two equal segments. (98)

Midsegment of a triangle: A line segment that connects the midpoints of two of a triangle’s sides. (286)

p. 733

Net: A two-dimensional pattern of polygons that, when folded together, forms a polyhedron. (635)

Nonagon: A polygon that has nine sides. (573)

Noncollinear points: Points that do not lie on the same line. (9)

Oblique line and plane (or two planes): A line and a plane (or two planes) that are neither parallel nor perpendicular. (620)

Obtuse angle: An angle whose measure is more than $90^\circ$ but less than $180^\circ$. (92)

Obtuse triangle: A triangle that has an obtuse angle. (158)

Octagon: A polygon that has eight sides. (19)

Octahedron: A polyhedron that has eight faces. The faces of a regular octahedron are equilateral triangles. (678)

Opposite rays: Informally, two rays are opposite rays if they have a common endpoint and point in opposite directions. Formally, rays AB and AC are opposite rays iff B-A-C. (19)

Orthocenter of a triangle: The point in which the lines containing the altitudes of a triangle are concurrent. (549)

Parallel line and plane: A line and a plane that do not intersect. (619)

Parallel lines: Lines that lie in the same plane and do not intersect. (118)

Parallel planes: Planes that do not intersect. (619)

Parallelogram: A quadrilateral whose opposite sides are parallel. (265)

Pentagon: A polygon that has five sides. (19)

Perimeter of a polygon: The sum of the lengths of the sides of a polygon. (20)

Perpendicular bisector of a line segment: The line that is perpendicular to the line segment and that divides the segment into two equal parts. (212)

Perpendicular line and plane: A line and a plane that intersect such that the line is perpendicular to every line in the plane that passes through the point of intersection. (620)

Perpendicular lines: Lines that form a right angle. (117)

Perpendicular planes: Two planes such that one plane contains a line that is perpendicular to the other plane. (620)

Plane: A flat unbounded surface. (9)

Polygon: A connected set of at least three line segments in the same plane such that each segment intersects exactly two others, one at each endpoint. (139)

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Polygonal region: The union of a polygon and its interior. (339)

Polyhedron: A solid bounded by parts of intersecting planes. (628)

Postulate: A statement that is assumed to be true without proof. We choose our postulates for their natural intuitive appeal and for their ability to capture the basic properties of space from which the rest of geometry can be developed. (61)

Prism: Suppose that $A$ and $B$ are two parallel planes, $R$ is a polygonal region in one plane, and l is a line that intersects both planes but not $R$. The solid made up of all segments parallel to line l that connect a point of region $R$ to a point of the other plane is a prism. (634)

Proportion: An equality between two ratios. (379)

Pyramid: Suppose that $A$ is a plane, $R$ is a polygonal region in plane $A$, and $P$ is a point not in plane $A$. The solid made up of all segments that connect $P$ to a point of region $R$ is a pyramid. Point $P$ is the apex of the pyramid. (647)

Pythagorean triple: A set of three integers that can be the lengths of the sides of a right triangle. (437)

Quadrilateral: A polygon that has four sides. (19)

Radius of a circle: (1) The distance between the center of a circle and any point on it. (2) A line segment that connects the center of the circle to any point on it. (485)

Radius of a regular polygon: A line segment that connects the center of the polygon to a vertex of the polygon. See Apothem of a regular polygon. (574)

Ratio: The ratio of the number $a$ to the number $b$ is the number $\dfrac{a}{b}$. (379)

Ray: Part of a line that extends endlessly in one direction. If a ruler is placed on a line, points with coordinates that are zero or positive form a ray. The ray in the figure below is named ray AB. (13)

Rectangle: A quadrilateral each of whose angles is a right angle. (259)

Rectangular solid: A polyhedron that has six rectangular faces. (628)

Reflection of a point through a line: The reflection of point P through line l is point P itself if P is on l. Otherwise, it is point P′ such that l is the perpendicular bisector of PP′. (305)

p. 735

Reflection symmetry: A figure has reflection (line) symmetry with respect to a line iff it coincides with its reflection image through the line. (320)

Reflex angle: An angle whose measure is more than $180^\circ$. (498)

Regular polygon: A convex polygon that is both equilateral and equiangular. (572)

Regular polyhedron: A convex solid having faces that are congruent regular polygons and having an equal number of polygons that meet at each vertex. Euclid showed that there are five regular polyhedra: the tetrahedron, the cube, the octahedron, the dodecahedron, and the icosahedron. (678)

Rhombus: A quadrilateral all of whose sides are equal. (276)

Right angle: An angle whose measure is $90^\circ$. (92)

Right triangle: A triangle that has a right angle. (158)

Rotation: Informally, a transformation in which a figure is rotated a certain number of degrees about a fixed point. Formally, a transformation that is the composite of two successive reflections through intersecting lines. (307)

Rotation symmetry: A figure has rotation symmetry with respect to a point iff it coincides with its rotation image through less than $360^\circ$ about the point. (319)

Saccheri quadrilateral: A birectangular quadrilateral whose legs are equal. (697)

Scalene triangle: A triangle that has no equal sides. (157)

Secant: A line that intersects a circle in two points. (511)

Secant angle: An angle whose sides are contained in two secants of a circle so that each side intersects the circle in at least one point other than the angle’s vertex. (511)

Sector of a circle: A region bounded by an arc of the circle and the two radii to the endpoints of the arc. (605)

Similar figures: Informally, two figures are similar if they have the same shape, though they may differ in size. In similar figures, corresponding segments are proportional and corresponding angles are equal. (386–387)

Similar solids: See Similar figures. (670)

p. 736

Similar triangles: Two triangles for which there is a correspondence between their vertices such that their corresponding sides are proportional and their corresponding angles are equal. (386)

Sine of an acute angle of a right triangle: The ratio of the length of the opposite leg to the length of the hypotenuse. More generally, see Trigonometric functions. (455)

Skew lines: Two lines that are not parallel and do not intersect. (619)

Slope: The slope $m$ of a nonvertical line that contains the points P$_1$($x_1$, $y_1$) and P$_2$($x_2$, $y_2$) is $m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}$. (462)

Sphere: The set of all points in space that are at a given distance from a given point. (662)

Square: A quadrilateral all of whose sides and angles are equal. (276)

Straight angle: An angle whose measure is $180^\circ$. (92)

Supplementary angles: Two angles whose sum is $180^\circ$. (106)

Symmetry with respect to a line: Two points are symmetric with respect to a line iff the line is the perpendicular bisector of the line segment connecting the two points. See Reflection symmetry. (212)

Symmetry with respect to a point: Two points are symmetric with respect to a point iff the point is the midpoint of the line segment connecting the two points. A figure that has symmetry with respect to a point coincides with its rotation image through $180^\circ$ about the point. (266)

Tangent: A line in the plane of a circle that intersects the circle in exactly one point. (491)

Tangent of an acute angle of a right triangle: The ratio of the length of the opposite leg to the length of the adjacent leg. More generally, see Trigonometric functions. (449)

Tangent segment: Any segment of a line that is tangent to a circle that has the point of tangency as one of the segment’s endpoints. (517)

Tetrahedron: A polyhedron that has four faces. The faces of a regular tetrahedron are equilateral triangles. (678)

p. 737

Theorem: A statement that is proved by reasoning deductively from already accepted statements. (51)

Transformation: A one-to-one correspondence between two sets of points. Examples of transformations include isometries, which preserve the size and shape of figures. Isometries give us the general definition of congruent figures. Isometries include reflections, translations, rotations, and glide reflections. Also important are dilations, which preserve shape but may alter the size of figures. (299–300)

Translation: Informally, a transformation in which a figure is slid a certain distance in a given direction without being turned. Formally, a transformation that is the composite of two successive reflections through parallel lines. (307)

Translation symmetry: A pattern has translation symmetry if it coincides with a translation image. (320)

Transversal: A line that intersects two or more lines that lie in the same plane in different points. (219)

Trapezoid: A quadrilateral that has exactly one pair of parallel sides. (281)

Triangle: A polygon that has three sides. (19)

Trigonometric functions: For acute angles, see Cosine, Sine, and Tangent. More generally, you can discover by working with a calculator that you can find $\cos x$, $\sin x$, and $\tan x$ where the number of degrees of $x$ ranges widely. The interpretation of these trigonometric functions is as follows. If a unit segment with one endpoint at the origin is rotated counterclockwise through an angle of $x$ degrees starting with its other endpoint at $(1, 0)$ on the $x$-axis, the other endpoint of this segment will end up at the point having coordinates $(\cos x, \sin x)$. Other trigonometric functions such as the tangent can be defined for general angles by the equations that define them in terms of sines and cosines of acute angles; for example, $\tan x=\dfrac{\sin x}{\cos x}$.

Vertical angles: Two angles such that the sides of one angle are opposite rays to the sides of the other. (111)

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Formulary

CIRCLE

Circumference: $c=\pi d$

$c=2\pi r$

Area: $A=\pi r^2$

Central angle: $\angle 1 = m\widehat{AB}$

Inscribed angle: $\angle 2 = \dfrac{1}{2}m\widehat{AB}$

Secant angles: $\angle 3 = \dfrac{1}{2}(m\widehat{AB}+m\widehat{CD})$

$\angle 4 = \dfrac{1}{2}(m\widehat{AB}-m\widehat{CD})$

DISTANCE AND SLOPE

Distance: $d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

Slope: $m=\dfrac{\text{rise}}{\text{run}}=\dfrac{y_2-y_1}{x_2-x_1}$

PARALLELOGRAM

Area: $A=bh$

PRISM

Volume: $V=Bh$

CONE (Circular)

Volume: $V=\dfrac{1}{3}Bh$

$V=\dfrac{1}{3}\pi r^2h$

PYRAMID

Volume: $V=\dfrac{1}{3}Bh$

CUBE

Diagonal: $d=e\sqrt{3}$

Area: $A=6e^2$

Volume: $V=e^3$

QUADRILATERAL (Convex)

Angle sum: $\angle A+\angle B+\angle C+\angle D=360^\circ$

CYLINDER (Right circular)

Lateral area: $A=2\pi rh$

Total area: $A=2\pi rh+2\pi r^2$

Volume: $V=Bh=\pi r^2h$

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RECTANGLE

Perimeter: $p=2l+2w$

Area: $A=lw$

RECTANGULAR SOLID

Diagonal: $d=\sqrt{l^2+w^2+h^2}$

Area: $A=2lw+2wh+2lh$

Volume: $V=lwh$

REGULAR POLYGON (n sides)

$N=n\sin\dfrac{180}{n}$

$M=n\sin\dfrac{180}{n}\cos\dfrac{180}{n}$

Perimeter: $p=ns=2Nr$

Area: $A=Mr^2$

SIMILAR SOLIDS

Given the edge ratio, $\dfrac{e_2}{e_1}=r$

Area ratio: $\dfrac{A_2}{A_1}=r^2$

Volume ratio: $\dfrac{V_2}{V_1}=r^3$

SIMILAR TRIANGLES (AND POLYGONS)

Given the side ratio, $\dfrac{s_2}{s_1}=r$

Altitude ratio: $\dfrac{h_2}{h_1}=r$

Perimeter ratio: $\dfrac{p_2}{p_1}=r$

Area ratio: $\dfrac{A_2}{A_1}=r^2$

SLOPE RELATIONS

Parallel lines: $m_1=m_2$

Perpendicular lines: $m_1m_2=-1$,

so $m_1=-\dfrac{1}{m_2}$

SPHERE

Area: $A=4\pi r^2$

Volume: $V=\dfrac{4}{3}\pi r^3$

SQUARE

Diagonal: $d=s\sqrt{2}$

Perimeter: $p=4s$

Area: $A=s^2$

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TRAPEZOID

Area: $A=\dfrac{1}{2}h(b_1+b_2)$

TRIANGLE

Perimeter: $p=a+b+c$

Area: $A=\dfrac{1}{2}bh$

Angle sum: $\angle A+\angle B+\angle C=180^\circ$

Law of sines: $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$

Law of cosines: $a^2=b^2+c^2-2bc\cos A$

$b^2=a^2+c^2-2ac\cos B$

$c^2=a^2+b^2-2ab\cos C$

Side-Splitter Theorem:

$\dfrac{a}{b}=\dfrac{c}{d}$

$\dfrac{a}{a+b}=\dfrac{c}{c+d}$

$\dfrac{b}{a+b}=\dfrac{d}{c+d}$

Ceva’s Theorem:

$\dfrac{a}{b}\cdot\dfrac{c}{d}\cdot\dfrac{e}{f}=1$

TRIANGLE, EQUILATERAL

Altitude: $h=\dfrac{\sqrt{3}}{2}s$

Area: $A=\dfrac{\sqrt{3}}{4}s^2$

TRIANGLE, RIGHT

Area: $A=\dfrac{1}{2}ab$

Pythagorean Theorem: $c^2=a^2+b^2$

Sine ratio: $\sin A=\dfrac{a}{c}$, $\sin B=\dfrac{b}{c}$

Cosine ratio: $\cos A=\dfrac{b}{c}$, $\cos B=\dfrac{a}{c}$

Tangent ratio: $\tan A=\dfrac{a}{b}$, $\tan B=\dfrac{b}{a}$

Altitude mean: $\dfrac{x}{h}=\dfrac{h}{y}$

Leg means: $\dfrac{c}{a}=\dfrac{a}{x}$, $\dfrac{c}{b}=\dfrac{b}{y}$

TRIANGLE, RIGHT (Isosceles)

Leg: $a$

Hypotenuse: $c=a\sqrt{2}$

TRIANGLE, RIGHT (30°–60°)

Shorter leg: $a$

Longer leg: $b=\sqrt{3}a=a\sqrt{3}$

Hypotenuse: $c=2a$

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Teacher: “. . . and now I want to prove this theorem.” Pupil: “Why bother to prove it, teacher? I take your word for it.”

Postulates and Theorems

Chapter 2

Postulate 1 Two points determine a line. (61)

Postulate 2 Three noncollinear points determine a plane. (61)

Theorem The Pythagorean Theorem. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. (65)

Theorem The Triangle Angle Sum Theorem. The sum of the angles of a triangle is $180^\circ$. (66)

Theorem If the diameter of a circle is $d$, its circumference is $\pi d$. (66)

Theorem If the radius of a circle is $r$, its area is $\pi r^2$. (66)

Chapter 3

Postulate 3 The Ruler Postulate. The points on a line can be numbered so that positive number differences measure distances. (85)

Postulate 4 The Protractor Postulate. The rays in a half-rotation can be numbered from 0 to 180 so that positive number differences measure angles. (92)

Theorem 1 The Betweenness of Points Theorem. If A-B-C, then AB + BC = AC. (86)

Theorem 2 The Betweenness of Rays Theorem. If OA-OB-OC, then ∠AOB + ∠BOC = ∠AOC. (93)

Corollary to the Ruler Postulate A line segment has exactly one midpoint. (99)

p. 742

Corollary to the Protractor Postulate An angle has exactly one ray that bisects it. (100)

Theorem 3 Complements of the same angle are equal. (106)

Theorem 4 Supplements of the same angle are equal. (106)

Theorem 5 The angles in a linear pair are supplementary. (111)

Theorem 6 Vertical angles are equal. (112)

Theorem 7 Perpendicular lines form right angles. (118)

Corollary to the definition of a right angle All right angles are equal. (118)

Theorem 8 If the angles in a linear pair are equal, then their sides are perpendicular. (118)

Chapter 4

The Distance Formula The distance between the points P$_1$($x_1$, $y_1$) and P$_2$($x_2$, $y_2$) is $\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$. (134)

Postulate 5 The ASA Postulate. If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent. (147)

Postulate 6 The SAS Postulate. If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. (147)

Corollary to the definition of congruent triangles Two triangles congruent to a third triangle are congruent to each other. (141)

Theorem 9 If two sides of a triangle are equal, the angles opposite them are equal. (158)

Corollary An equilateral triangle is equiangular. (159)

Theorem 10 If two angles of a triangle are equal, the sides opposite them are equal. (159)

Corollary An equiangular triangle is equilateral. (159)

Theorem 11 The SSS Theorem. If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. (164)

Chapter 5

Theorem 12 The Exterior Angle Theorem. An exterior angle of a triangle is greater than either remote interior angle. (191)

Theorem 13 If two sides of a triangle are unequal, the angles opposite them are unequal in the same order. (196)

Theorem 14 If two angles of a triangle are unequal, the sides opposite them are unequal in the same order. (196)

Theorem 15 The Triangle Inequality Theorem. The sum of any two sides of a triangle is greater than the third side. (201)

Chapter 6

Postulate 7 The Parallel Postulate. Through a point not on a line, there is exactly one line parallel to the given line. (226)

Theorem 16 In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment. (212)

Theorem 17 Equal corresponding angles mean that lines are parallel. (220)

Corollary 1 Equal alternate interior angles mean that lines are parallel. (220)

Corollary 2 Supplementary interior angles on the same side of a transversal mean that lines are parallel. (220)

Corollary 3 In a plane, two lines perpendicular to a third line are parallel. (220)

Theorem 18 In a plane, two lines parallel to a third line are parallel to each other. (226)

Theorem 19 Parallel lines form equal corresponding angles. (231)

Corollary 1 Parallel lines form equal alternate interior angles. (231)

p. 743

Corollary 2 Parallel lines form supplementary interior angles on the same side of a transversal. (231)

Corollary 3 In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other. (231)

Theorem 20 The Angle Sum Theorem. The sum of the angles of a triangle is $180^\circ$. (237)

Corollary 1 If two angles of one triangle are equal to two angles of another triangle, the third angles are equal. (237)

Corollary 2 The acute angles of a right triangle are complementary. (237)

Corollary 3 Each angle of an equilateral triangle is $60^\circ$. (237)

Theorem 21 An exterior angle of a triangle is equal to the sum of the remote interior angles. (237)

Theorem 22 The AAS Theorem. If two angles and the side opposite one of them in one triangle are equal to the corresponding parts of another triangle, the triangles are congruent. (243)

Theorem 23 The HL Theorem. If the hypotenuse and a leg of one right triangle are equal to the corresponding parts of another right triangle, the triangles are congruent. (244)

Chapter 7

Theorem 24 The sum of the angles of a quadrilateral is $360^\circ$. (259)

Corollary A quadrilateral is equiangular iff it is a rectangle. (259)

Theorem 25 The opposite sides and angles of a parallelogram are equal. (266)

Theorem 26 The diagonals of a parallelogram bisect each other. (266)

Theorem 27 A quadrilateral is a parallelogram if its opposite sides are equal. (271)

Theorem 28 A quadrilateral is a parallelogram if its opposite angles are equal. (271)

Theorem 29 A quadrilateral is a parallelogram if two opposite sides are both parallel and equal. (271)

Theorem 30 A quadrilateral is a parallelogram if its diagonals bisect each other. (271)

Theorem 31 All rectangles are parallelograms. (276)

Theorem 32 All rhombuses are parallelograms. (276)

Theorem 33 The diagonals of a rectangle are equal. (277)

Theorem 34 The diagonals of a rhombus are perpendicular. (277)

Theorem 35 The base angles of an isosceles trapezoid are equal. (282)

Theorem 36 The diagonals of an isosceles trapezoid are equal. (282)

Theorem 37 The Midsegment Theorem. A midsegment of a triangle is parallel to the third side and half as long. (287)

Chapter 9

Postulate 8 The Area Postulate. Every polygonal region has a positive number called its area such that (1) congruent triangles have equal areas and (2) the area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts. (339)

Postulate 9 The area of a rectangle is the product of its base and altitude. (345)

Corollary to Postulate 9 The area of a square is the square of its side. (345)

Theorem 38 The area of a right triangle is half the product of its legs. (352)

Theorem 39 The area of a triangle is half the product of any base and corresponding altitude. (353)

Corollary Triangles with equal bases and equal altitudes have equal areas. (353)

Theorem 40 The area of a parallelogram is the product of any base and corresponding altitude. (359)

p. 744

Theorem 41 The area of a trapezoid is half the product of its altitude and the sum of its bases. (359)

Theorem 42 The Pythagorean Theorem. The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs. (366)

Theorem 43 Converse of the Pythagorean Theorem. If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is a right triangle. (366)

Chapter 10

Theorem 44 The Side-Splitter Theorem. If a line parallel to one side of a triangle intersects the other two sides in different points, it divides the sides in the same ratio. (393)

Corollary If a line parallel to one side of a triangle intersects the other two sides in different points, it cuts off segments proportional to the sides. (394)

Theorem 45 The AA Theorem. If two angles of one triangle are equal to two angles of another triangle, the triangles are similar. (400)

Corollary Two triangles similar to a third triangle are similar to each other. (401)

Theorem 46 Corresponding altitudes of similar triangles have the same ratio as that of the corresponding sides. (408)

Theorem The SAS Similarity Theorem. If an angle of one triangle is equal to an angle of another triangle and the sides including these angles are proportional, then the triangles are similar. (412)

Theorem The SSS Similarity Theorem. If the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar. (412)

Theorem 47 The ratio of the perimeters of two similar polygons is equal to the ratio of the corresponding sides. (415)

Theorem 48 The ratio of the areas of two similar polygons is equal to the square of the ratio of the corresponding sides. (416)

Chapter 11

Theorem 49 The altitude to the hypotenuse of a right triangle forms two triangles similar to it and to each other. (429)

Corollary 1 The altitude to the hypotenuse of a right triangle is the geometric mean between the segments into which it divides the hypotenuse. (429)

Corollary 2 Each leg of a right triangle is the geometric mean between the hypotenuse and its projection on the hypotenuse. (429)

Theorem 50 The Isosceles Right Triangle Theorem. In an isosceles right triangle, the hypotenuse is $\sqrt{2}$ times the length of a leg. (442)

Corollary Each diagonal of a square is $\sqrt{2}$ times the length of one side. (442)

Theorem 51 The 30°-60° Right Triangle Theorem. In a 30°-60° right triangle, the hypotenuse is twice the shorter leg and the longer leg is $\sqrt{3}$ times the shorter leg. (442)

Corollary An altitude of an equilateral triangle having side $s$ is $\dfrac{\sqrt{3}}{2}s$ and its area is $\dfrac{\sqrt{3}}{4}s^2$. (443)

Theorem 52 Two nonvertical lines are parallel iff their slopes are equal. (463)

Theorem 53 Two nonvertical lines are perpendicular iff the product of their slopes is $-1$. (463)

Theorem 54 The Law of Sines. If the sides opposite ∠A, ∠B, and ∠C of △ABC have lengths $a$, $b$, and $c$, then $\dfrac{\sin A}{a}=\dfrac{\sin B}{b}=\dfrac{\sin C}{c}$. (469)

Theorem 55 The Law of Cosines. If the sides opposite ∠A, ∠B, and ∠C of △ABC have lengths $a$, $b$, and $c$, then $c^2=a^2+b^2-2ab\cos C$. (469)

p. 745

Chapter 12

Postulate 10 The Arc Addition Postulate. If C is on $\widehat{AB}$, then $m\widehat{AC}+m\widehat{CB}=m\widehat{ACB}$. (499)

Corollary to the definition of a circle All radii of a circle are equal. (485)

Theorem 56 If a line through the center of a circle is perpendicular to a chord, it also bisects the chord. (485)

Theorem 57 If a line through the center of a circle bisects a chord that is not a diameter, it is also perpendicular to the chord. (485)

Theorem 58 The perpendicular bisector of a chord of a circle contains the center of the circle. (485)

Theorem 59 If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of contact. (492)

Theorem 60 If a line is perpendicular to a radius at its outer endpoint, it is tangent to the circle. (492)

Theorem 61 In a circle, equal chords have equal arcs. (499)

Theorem 62 In a circle, equal arcs have equal chords. (499)

Theorem 63 An inscribed angle is equal in measure to half its intercepted arc. (505)

Corollary 1 Inscribed angles that intercept the same arc are equal. (505)

Corollary 2 An angle inscribed in a semicircle is a right angle. (505)

Theorem 64 A secant angle whose vertex is inside a circle is equal in measure to half the sum of the arcs intercepted by it and its vertical angle. (511)

Theorem 65 A secant angle whose vertex is outside a circle is equal in measure to half the difference of its larger and smaller intercepted arcs. (511)

Theorem 66 The Tangent Segments Theorem. The tangent segments to a circle from an external point are equal. (517)

Theorem 67 The Intersecting Chords Theorem. If two chords intersect in a circle, the product of the lengths of the segments of one chord is equal to the product of the lengths of the segments of the other chord. (517)

Chapter 13

Theorem 68 Every triangle is cyclic. (531)

Corollary The perpendicular bisectors of the sides of a triangle are concurrent. (531)

Theorem 69 A quadrilateral is cyclic iff a pair of its opposite angles are supplementary. (536)

Theorem 70 Every triangle has an incircle. (543)

Corollary The angle bisectors of a triangle are concurrent. (543)

Theorem 71 The medians of a triangle are concurrent. (549)

Theorem 72 The lines containing the altitudes of a triangle are concurrent. (549)

Theorem 73 Ceva’s Theorem. Three cevians, AY, BZ, and CX, of △ABC are concurrent iff $\dfrac{AX}{XB}\cdot\dfrac{BY}{YC}\cdot\dfrac{CZ}{ZA}=1$. (555)

Chapter 14

Theorem 74 Every regular polygon is cyclic. (573)

Theorem 75 The perimeter of a regular polygon having $n$ sides is $2Nr$, in which $N=n\sin\dfrac{180}{n}$ and $r$ is its radius. (581)

Theorem 76 The area of a regular polygon having $n$ sides is $Mr^2$, in which $M=n\sin\dfrac{180}{n}\cos\dfrac{180}{n}$ and $r$ is its radius. (587)

Theorem 77 If the radius of a circle is $r$, its circumference is $2\pi r$. (592)

Corollary If the diameter of a circle is $d$, its circumference is $\pi d$. (592)

Theorem 78 If the radius of a circle is $r$, its area is $\pi r^2$. (599)

p. 746

Chapter 15

Postulate 11 If two points lie in a plane, the line that contains them lies in the plane. (619)

Postulate 12 If two planes intersect, they intersect in a line. (619)

Postulate 13 Cavalieri’s Principle. Consider two geometric solids and a plane. If every plane parallel to this plane that intersects one of the solids also intersects the other so that the resulting cross sections have equal areas, then the two solids have equal volumes. (641)

Postulate 14 The volume of any prism is the product of the area of its base and its altitude: $V=Bh$. (642)

Theorem 79 The length of a diagonal of a rectangular solid with dimensions $l$, $w$, and $h$ is $\sqrt{l^2+w^2+h^2}$. (629)

Corollary The length of a diagonal of a cube with edges of length $e$ is $e\sqrt{3}$. (629)

Corollary 1 to Postulate 14 The volume of a rectangular solid is the product of its length, width, and height: $V=lwh$. (642)

Corollary 2 to Postulate 14 The volume of a cube is the cube of its edge: $V=e^3$. (642)

Theorem 80 The volume of any pyramid is one-third of the product of the area of its base and its altitude: $V=\dfrac{1}{3}Bh$. (648)

Theorem 81 The volume of a cylinder is the product of the area of its base and its altitude: $V=Bh=\pi r^2h$. (655)

Theorem 82 The volume of a cone is one-third of the product of the area of its base and its altitude: $V=\dfrac{1}{3}Bh=\dfrac{1}{3}\pi r^2h$. (656)

Theorem 83 The volume of a sphere is $\dfrac{4}{3}\pi$ times the cube of its radius: $V=\dfrac{4}{3}\pi r^3$. (664)

Theorem 84 The surface area of a sphere is $4\pi$ times the square of its radius: $V=4\pi r^2$. (665)

Theorem 85 If the ratio of a pair of corresponding dimensions of two similar solids is $r$, then the ratio of their surface areas is $r^2$. (672)

Theorem 86 If the ratio of a pair of corresponding dimensions of two similar solids is $r$, then the ratio of their volumes is $r^3$. (672)

Chapter 16

Theorem 87 The summit angles of a Saccheri quadrilateral are equal. (698)

Theorem 88 The line segment connecting the midpoints of the base and summit of a Saccheri quadrilateral is perpendicular to both of them. (698)

Theorem 89 If the legs of a birectangular quadrilateral are unequal, the summit angles opposite them are unequal in the same order. (698)

Theorem 90 If the summit angles of a birectangular quadrilateral are unequal, the legs opposite them are unequal in the same order. (698)

The Lobachevskian Postulate The summit angles of a Saccheri quadrilateral are acute. (703)

Lobachevskian Theorem 1 The summit of a Saccheri quadrilateral is longer than its base. (703)

Lobachevskian Theorem 2 A midsegment of a triangle is less than half as long as the third side. (703)

Lobachevskian Theorem 3 The sum of the angles of a triangle is less than $180^\circ$. (708)

Corollary The sum of the angles of a quadrilateral is less than $360^\circ$. (708)

Lobachevskian Theorem 4 If two triangles are similar, they must also be congruent. (708)

p. 747

Answers to Selected Exercises

Chapter 1, Lesson 1 (pages 10–11)

2. 12 cm.

5. About 1.3 cm.

7. 3 mi.

14. Line segments.

21. Three.

25. Six; AB, AG, AC, AF, AD, AI.

Chapter 1, Lesson 2 (pages 15–17)

2. ∠C.

6. Rays AC and AB.

10. ∠B = 30°.

13. Collinear.

16. ∠1 = 67°.

22. ∠I = 117°.

25. Approximately $1\dfrac{3}{4}$ ft.

28. Approximately $3\dfrac{1}{4}$ ft.

30. 34°.

Chapter 1, Lesson 3 (pages 21–22)

2. 9 in².

3. 12 in.

4. 36 in².

5. 24 in.

13. Piece I because it has four sides.

16. Ranch A.

23. Polygons.

25. Eight.

28. Three.

30. A line.

32. 75.6 ft.

Chapter 1, Lesson 4 (pages 26–29)

5. Noncollinear.

11. The bisecting lines seem to be concurrent.

16. No.

17. West.

21. About 6 A.M. and 6 P.M.

23. OI bisects ∠HOJ.

25. ∠DOB and ∠EOA.

31. 60°.

33. 180°.

Chapter 1, Lesson 5 (pages 31–34)

1. 100 square units.

6. It contains 80 square units.

8. 54 square units.

11. It contains 84 square units.

Chapter 1, Review (pages 36–38)

2. Polygons (or squares).

5. 12.

8. Coplanar.

10. PQ.

13. T.

18. Constructions.

20. ∠DAB = 110°.

25. ∠DCF = ∠FCE = 40°.

Chapter 1, Algebra Review (page 40)

1. Associative for addition.

2. Commutative for multiplication.

3. Definition of subtraction.

4. Identity for addition.

5. Definition of division.

6. Inverse for multiplication.

7. 1,728.

8. 12,000.

9. 20.

10. −75.

11. 225.

12. 81.

13. $49\pi$.

14. $14\pi$.

15. $5x$.

16. $x^5$.

17. $5x$.

18. $x^5$.

19. $2x+3y$.

20. $x^2y^3$.

21. $x^2+y^3$.

22. $x^3+y^2$.

23. $x+2y$.

24. $xy^2$.

25. $7x^2$.

26. $7x^2$.

27. $5x^2$.

28. $5x^2$.

29. $4x^2+3x$.

30. $3x^3+2x^2+x$.

31. $x^{14}$.

32. $x^{11}$.

33. $6x^5$.

34. $6x^5$.

35. $36x^2$.

36. $72x^5$.

37. $4x+4y$.

38. $4x+4y$.

39. $2x$.

40. $2x$.

41. 0.

42. 1.

43. $5x+10y$.

44. $3x+4y$.

45. $5x-10y$.

46. $3x-4y$.

47. $31xy$.

48. $25xy$.

49. $7+x^3+x^5$.

50. $10x^8$.

Chapter 2, Lesson 1 (pages 43–45)

1. A conditional statement.

4. You live in the Ozarks.

6. Yes.

14. If an animal is a koala bear, then it eats only eucalyptus leaves.

16. If I don’t understand, then I ask questions.

21. If at first you don’t succeed, then try again.

24. Regions 1 and 2.

28. The second.

35. Statements c and d.

Chapter 2, Lesson 2 (pages 47–49)

3. If a creature has two legs, then it is a goose.

5. You are not more than six feet tall.

9. If you do not know how to reason deductively.

13. They have the same meaning.

21. Yes. The converse must be true because this is a definition.

26. No.

28. It is the converse.

32. Sentence 2.

35. It is the converse.

Chapter 2, Lesson 3 (pages 52–54)

2. If you keep quiet, others will think you are wise.

4. All Greeks are humans. All humans are mortals. Therefore, all Greeks are mortals.

7. No.

8. The hypothesis.

12. It also may be false.

16. A theorem.

18. If you go to Dallas, you will see the yellow rows of taxis.

22. 21.

26. Red. Because the label is wrong.

Chapter 2, Lesson 4 (pages 57–59)

1. Contradiction.

3. Suppose that it would not speak foul language.

6. No.

10. It is the opposite of the theorem’s conclusion.

11. The earth is not flat.

14. That it is false.

p. 748

15. Beginning assumption: Suppose that the pupils can obey the teacher. Contradiction: There are 13 pupils at the black desks but only 12 brown desks. Conclusion: The pupils can’t obey the teacher.

Chapter 2, Lesson 5 (pages 61–64)

3. You go around in circles.

6. Two points determine a line.

8. Theorems.

12. The points.

18. Converse.

20. True.

22. False.

26. A picul is 1,600 taels.

32. Statements 1 and 3.

37. Two points determine a line.

Chapter 2, Lesson 6 (pages 67–70)

1. The length of the hypotenuse of a right triangle.

5. The area of a circle.

9. No.

12. $2\pi$, $4\pi$, $6\pi$, $8\pi$, and $10\pi$ units.

13. $9\pi$ square units.

18. 30°.

21. 70°.

24. No.

26. Approximately 50 mm².

32. Yes, because the sum of the other two angles could be 1°.

34. $a^2$.

38. To the left.

Chapter 2, Review (pages 71–74)

1. If something is a limerick, then it has five lines.

9. Collinear.

11. A postulate.

13. If it rains, then it pours.

18. 50°.

20. Three noncollinear points determine a plane.

23. Square.

32. Theorems.

34. 144 square units.

37. 20 ft.

41. Direct.

44. $2\pi r$.

48. $4r^2$.

Chapter 2, Algebra Review (page 76)

1. $10x$.

2. $16x^2$.

3. $6x+5$.

4. $4x-5$.

5. $6x^2$.

6. $-6x^2$.

7. $9x+4y$.

8. $3x+2y$.

9. $5x+15$.

10. $44-11x$.

11. $18x+6$.

12. $40-56x$.

13. $9x^2-18x$.

14. $20x+6x^2$.

15. $x^2y+xy^2$.

16. $2x^2-4x+8$.

17. 10.

18. 8.

19. 3.

20. 6.

21. 13.

22. 9.

23. −7.

24. −5.

25. 60.

26. 0.

27. −12.

28. 37.

29. −2.

30. 4.5.

31. 12.

32. Any number.

Chapter 3, Lesson 1 (pages 80–82)

1. Subtraction.

2. Substitution.

10. Addition.

14. No number (because there is no number when multiplied by 0 that gives 6).

19. If $a=b$ and $c\neq0$, then $\dfrac{a}{c}=\dfrac{b}{c}$. If $c=d$, then $\dfrac{a}{c}=\dfrac{b}{d}$ by substitution.

21. Substitution.

25. $a^2$ and $b^2$.

26. $2ab$.

28. $(a+b)^2$.

32. Substitution.

38. Division.

41. A direct proof.

Chapter 3, Lesson 2 (pages 86–90)

4. The definition of betweenness of points.

7. 52.

9. BC + CD = BD, or 5 + 6 = 11.

15. 92,760,000 mi.

17. 18.

20. 9 − 2 = 7.

26. A, −25; B, 15.

30. B.

35. AB + BC is greater than AC.

38. AC = AB + BC.

Chapter 3, Lesson 3 (pages 93–97)

6. The definition of betweenness of rays.

8. Rays.

11. The coordinates.

13. Acute.

16. 15.

22. Obtuse.

25. 14.4°.

26. Yes.

29. 28.8°.

31. ∠SCP + ∠PCK = ∠SCK.

35. 27.

38. They are the same.

41. Betweenness of Rays Theorem.

44. 20°.

50. 360.

51. 15.

Chapter 3, Lesson 4 (pages 100–104)

3. Figures that can be made to coincide (or fit exactly together).

7. Then it divides it into two equal segments.

9. Each is the converse of the other.

15. No.

19. 45°.

22. 67.5°.

26. The midpoint of a line segment divides it into two equal segments.

28. 1.5.

31. 2 cm.

32. $2\pi$ (or about 6.28).

42. 180.

44. They are each 36°.

Chapter 3, Lesson 5 (pages 107–109)

1. 90°.

4. 87°.

8. 30°.

9. 150°.

16. ∠DOW + ∠WOF = ∠DOF.

19. 27°.

26. Betweenness of Rays Theorem.

28. Substitution.

35. 75°.

37. ∠S seems to be 90° larger than ∠C.

42. They are supplementary.

Chapter 3, Lesson 6 (pages 112–115)

1. Because they are vertical angles.

7. They are a linear pair and they are supplementary.

12. 80°.

14. They are vertical angles.

16. No.

23. Two.

26. 180°.

30. These angles are linear pairs; so they are supplementary.

36. Vertical angles are equal.

37. ∠3 = ∠1 + 90°.

45. 179°59′58.25″.

p. 749

Chapter 3, Lesson 7 (pages 119–121)

1. Two angles are complementary if their sum is 90°.

4. Supplements of the same angle are equal.

6. If the angles in a linear pair are equal, then their sides are perpendicular.

10. Vertical angles are equal.

14. DH, AE, and, if perspective is taken into account, HG, EF.

17. They are a linear pair.

21. They are perpendicular.

24. ∠1 and ∠3 look like vertical angles, yet they are not equal.

30. ∠BOX, 25°; ∠BOY, 65°.

33. 180°.

37. They are perpendicular.

Chapter 3, Review (pages 124–128)

1. Yes. E-F-H because 6.1 < 7.7 < 10.8.

5. $(180-x)^\circ$.

8. The handle is perpendicular to the blade.

11. CE-CS-CD.

13. 41.5.

18. 110.

23. They would be perpendicular.

28. Any number is equal to itself.

34. $(6+x)$ ft.

38. ∠AOB = $(90-x+y)^\circ$.

43. Hexagons.

46. Six.

50. 150.

53. 120 steps per minute.

56. 6,480 inches per minute.

Chapter 3, Algebra Review (page 130)

1. $6x+1$.

2. $2x-4y$.

3. $4x^2+8$.

4. $3x^2+6x-5$.

5. $12x-3$.

6. $x^3+4x^2-x-8$.

7. $14x-5y$.

8. $7x+4$.

9. $6x$.

10. $-x^2-11x$.

11. $2x^2+10x-1$.

12. $2x^3+y^3$.

13. $x^2+14x+24$.

14. $6x^2-7x-20$.

15. $16x^2-1$.

16. $x^3-10x^2-23x+6$.

17. $4y^3-8y^2+13y-5$.

18. $x^3+y^3$.

19. $5x^3+3x^2-20x-12$.

20. $x^3+5x^2-12x-36$.

21. −49.

22. $100x^2-20x+1$.

23. $x^3+12x^2+48x+64$.

24. $-4x+5$.

Chapter 4, Lesson 1 (pages 134–138)

3. A and E.

5. B.

7. A(4, 3), C(−3, −4).

9. On the y-axis.

12. Two.

16. The fourth (IV).

19. The hummingbirds.

23. Point A.

27. A(4, 1), B(6, 6). AB = $\sqrt{29}\approx5.39$.

31. A$_1$(5, 2), A$_2$(15, 2).

35. (900, 200).

37. 250 mm.

41. $\sqrt{18}$ or $3\sqrt{2}$.

Chapter 4, Lesson 2 (pages 141–144)

2. The vertices.

4. A 32-gon.

6. That they are congruent.

8. They are right angles (and equal).

11. Line segments AB and EF intersect two others but not at their endpoints.

12. △AEG.

17. CBEG.

25. An octagon (an 8-gon).

27. D.

31. H.

34. HBCFIE ↔︎ MNIEHK.

40. Convex.

45. No.

48. 21 square units.

53. No.

Chapter 4, Lesson 3 (pages 148–150)

1. BF and BP.

3. ∠B.

5. SAS.

10. TR = AN.

11. ∠I = ∠G.

15. Two triangles congruent to a third triangle are congruent to each other.

17. All the angles are acute.

20. None of the sides are equal.

22. All three.

29. 35°.

31. They get larger.

34. About 5.9 cm.

38. Perpendicular lines form right angles.

41. ASA.

44. Both are pairs of opposite rays.

Chapter 4, Lesson 4 (pages 153–155)

2. The midpoint of a line segment divides it into two equal segments.

7. Corresponding parts of congruent triangles are equal.

10. ASA.

17. If the angles in a linear pair are equal, then their sides are perpendicular.

19. Hexagons.

20. △ABX ≅ △CBX.

26. No. △BDC is larger than △ADB.

35. AB = 7, AC = 4, BC = $\sqrt{65}\approx8.1$.

37. △ABC and △DEF would coincide.

38. The x-coordinates are equal; the y-coordinates are opposites.

40. SAS.

44. Yes.

Chapter 4, Lesson 5 (pages 159–162)

3. ∠ABD and ∠ADB.

4. BD.

6. No.

8. If two sides of a triangle are equal, the angles opposite them are equal.

12. That it has at least two equal sides.

21. Addition.

22. Betweenness of Rays Theorem.

23. Substitution.

28. They appear to be parallel.

30. ∠S = ∠B = 72°.

38. 92.2°.

39. Obtuse.

43. A pentagon.

44. Because its sides intersect in points other than their endpoints.

50. They seem to be congruent.

52. It seems to be equilateral (and equiangular).

p. 750

Chapter 4, Lesson 6 (pages 165–167)

1. SSS.

2. Corresponding parts of congruent triangles are equal.

5. They are a linear pair.

10. Five.

11. No.

16. CA = CB and AD = BD.

18. SSS.

24. SAS.

29. ∠ADC = ∠CBA because they are corresponding parts of congruent triangles.

38. SSS.

40. OC bisects ∠AOB.

42. Because an angle has exactly one ray that bisects it (the corollary to the Protractor Postulate).

Chapter 4, Lesson 7 (pages 172–175)

1. A compass.

2. Distances between pairs of points.

5. A line segment is bisected.

11. ∠2 is complementary to ∠1.

12. Copying an angle.

18. It means equally distant.

25. Addition.

26. Substitution.

36. SSS.

37. Height = 7.5 in; width = 10 in.

Chapter 4, Review (pages 176–180)

2. Because they have the same size and shape (they can be made to coincide).

4. They are all equal.

8. Statement 4.

13. An equilateral triangle is equiangular.

15. Substitution.

17. 60°.

19. 120°.

21. 30°.

25. SSS.

29. 90°.

33. 45°.

38. $\sqrt{4+16}=\sqrt{20}$ (or $2\sqrt{5}\approx4.5$).

40. At (−5, 3).

46. Seven.

Chapter 4, Algebra Review (page 182)

1. $2(3x-1)$.

2. $3(2x-1)$.

3. $2(3x-2)$.

4. $4(x+3y)$.

5. $x(x+10)$.

6. $x^2(3x-2)$.

7. $5(3\pi-8)$.

8. $2x(4+y)$.

9. $x(a+b-c)$.

10. $2\pi r(h+r)$.

11. $(x+4)(x+4)$ or $(x+4)^2$.

12. $(x+4)(x-4)$.

13. $(x+3)(x+10)$.

14. $(x-2)(x+15)$.

15. $(x+7)(x-6)$.

16. $(2x-1)(x-7)$.

17. $(3x-5)(x+2)$.

18. $x(x-4y)$.

19. $(x-2y)(x-2y)$ or $(x-2y)^2$.

20. $(x+2y)(x-2y)$.

21. $\pi(a+r)(a-r)$.

22. $5x(x+2)(x-2)$.

23. $x(x+4)(x-3)$.

24. $z(2x-3)$.

25. $(x+2)(2x-3)$.

Chapter 5, Lesson 1 (pages 186–189)

1. Either $h>w$, $h=w$, or $h<w$.

6. The transitive property.

8. Multiplication.

11. Addition.

14. AB > BC.

15. Substitution.

20. ∠APB > ∠HPI.

22. ∠PIH < ∠PBA.

26. Transitive.

30. The “whole greater than part” theorem.

34. ∠A ≈ 83°, ∠B ≈ 56°, ∠C ≈ 41°.

37. 9 units.

41. That the line is “upright” (contains the center of the earth).

Chapter 5, Lesson 2 (pages 192–194)

1. ∠BCY, ∠BAX, ∠CAD (∠YAD).

3. It gets smaller.

6. An exterior angle of a triangle is greater than either remote interior angle.

9. ∠2, ∠5, ∠8.

10. Two.

16. Two.

19. ∠ROS.

20. ∠ROA > ∠SRO and ∠ROA > ∠S.

25. No. It does not form a linear pair with an angle of the triangle.

26. 360°.

28. 180°.

32. A line segment has exactly one midpoint.

37. The “whole greater than part” theorem.

39. Two points determine a line.

41. The angles in a linear pair are supplementary.

43. An exterior angle of a triangle is greater than either remote interior angle.

Chapter 5, Lesson 3 (pages 197–199)

3. Each is the converse of the other.

4. BC < AC.

8. ∠A > ∠B.

10. If two sides of a triangle are unequal, the angles opposite them are unequal in the same order.

14. ∠B.

18. ∠A ≈ 29°, ∠B ≈ 104°, ∠C ≈ 47°.

19. DE.

22. DF ≈ 4.5 cm, EF ≈ 5.4 cm.

23. ∠X > ∠Y.

30. ∠BDE.

34. BE bisects ∠ABC.

36. Yes. They are congruent by ASA.

42. FG < GH because both are sides of △FGH in which GH is the longest side.

45. An equilateral triangle is equiangular.

47. The “whole greater than part” theorem.

Chapter 5, Lesson 4 (pages 202–204)

2. DP + PH > DH.

6. They are collinear.

7. Yes; 160 million miles.

10. PA + PB > 12, PA + PC > 12, and PB + PC > 12.

17. A, B, and C are not collinear.

19. AB + BC = AC.

24. An angle has exactly one line that bisects it.

27. Substitution.

28. If two angles of a triangle are unequal, the sides opposite them are unequal in the same order.

33. The sum of any two sides of a triangle is greater than the third side.

35. Substitution.

44. 5.5 m.

p. 751

Chapter 5, Review (pages 206–209)

5. The “three possibilities” property.

7. Both dimensions are 2.5 in.

12. One angle has a measure of 179°, and the other two angles are each less than 1°.

16. The midpoint of a line segment divides it into two equal segments.

17. Reflexive.

20. If the angles in a linear pair are equal, their sides are perpendicular.

22. If two sides of a triangle are unequal, the angles opposite them are unequal in the same order.

24. Transitive.

26. The small angles each appear to be equal to 10°.

29. An exterior angle of a triangle is greater than either remote interior angle.

34. $n>2$.

37. One.

39. Yes.

41. SSS.

43. Betweenness of Rays Theorem.

Chapter 5, Algebra Review (page 210)

1. $\dfrac{4}{5}$.

2. $\dfrac{3x}{8}$.

3. $\dfrac{1}{x}$.

4. $\dfrac{x+1}{2}$.

5. $\dfrac{4}{7}$.

6. $\dfrac{1}{3}$.

7. $\dfrac{1}{x-5}$.

8. $\dfrac{x}{y}$.

9. $\dfrac{x+3}{10}$.

10. $\dfrac{x-4}{x-6}$.

11. $\dfrac{4}{4x}$.

12. $\dfrac{x^3}{x^4}$.

13. $\dfrac{-3}{x-2}$.

14. $\dfrac{8}{12}$ and $\dfrac{9}{12}$.

15. $\dfrac{28}{7x}$ and $\dfrac{2}{7x}$.

16. $\dfrac{y}{xy}$ and $\dfrac{x}{xy}$.

17. $\dfrac{3x}{3x+6}$ and $\dfrac{2}{3x+6}$.

18. $\dfrac{5(x+1)}{(x-1)(x+1)}$ or $\dfrac{5x+5}{x^2-1}$ and $\dfrac{x}{(x-1)(x+1)}$ or $\dfrac{x}{x^2-1}$.

19. $\dfrac{x^2}{xyz}$ and $\dfrac{y^2}{xyz}$.

Chapter 6, Lesson 1 (pages 214–217)

4. Yes, because W and E are equidistant from the endpoints of NS.

12. 6 cm.

13. 90 ft.

15. In the west.

19. Seven.

21. Line l is the perpendicular bisector of ST.

22. If a line segment is bisected, it is divided into two equal segments.

31. 9.5 in.

34. C and E; D and F.

36. Because they are corresponding parts of congruent triangles (△ABC ≅ △ADC by SSS).

39. N.

43. Three.

Chapter 6, Lesson 2 (pages 221–224)

1. (1) ∠1 = ∠2. (2) ∠2 = ∠3. (3) ∠1 = ∠3. (4) $a\parallel b$.

4. Alternate interior angles.

6. In a plane, two lines perpendicular to a third line are parallel.

7. Corresponding angles.

9. Equal corresponding angles mean that lines are parallel.

14. No.

18. $AE\parallel BF$.

27. No. These angles are not formed by a transversal. They could be equal and yet lines $x$ and $y$ could intersect either to the left or to the right.

32. Four.

35. SAS.

38. No.

Chapter 6, Lesson 3 (pages 226–229)

1. 12.

2. If they form a right angle.

5. ∠3.

6. Equal corresponding angles mean that lines are parallel.

12. At least, no more than, exactly.

14. No more than.

20. They form equal corresponding angles with a transversal.

22. They seem to be supplementary.

25. All four of them.

27. Down, below the figure.

30. One.

33. In a plane, two lines perpendicular to a third line are parallel.

36. Through a point not on a line, there is exactly one line parallel to the line.

37. Yes. (SAS.)

42. $a$ and $b$ are not parallel.

43. Given.

Chapter 6, Lesson 4 (pages 232–235)

1. They are equal.

2. Parallel lines form equal corresponding angles.

5. (1) Given. (2) Parallel lines form equal corresponding angles. (3) Vertical angles are equal. (4) Substitution.

9. Betweenness of Rays Theorem.

12. Equal corresponding angles mean that lines are parallel.

13. ∠1 = 115°, ∠2 = 65°, ∠3 = 115°.

17. They must be perpendicular because, in a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.

20. 67°, 113°, 67°, and 113°.

22. ∠1 and ∠2 are supplementary because parallel lines form supplementary interior angles on the same side of a transversal.

24. ∠3 = ∠4 because parallel lines form equal alternate interior angles.

30. Through a point not on a line, there is exactly one line parallel to the line (the Parallel Postulate).

31. Parallel lines form equal alternate interior angles.

37. Two points determine a line.

39. In a plane, two lines perpendicular to a third line are parallel.

p. 752

44. The length of a perpendicular segment between them.

46. 70°.

Chapter 6, Lesson 5 (pages 238–241)

1. The sum of the angles of a triangle is 180°.

2. Substitution.

3. Substitution.

4. Subtraction.

9. An equilateral triangle is equiangular.

10. An equiangular triangle has three equal angles.

14. Through a point not on a line, there is exactly one line parallel to the line.

16. Parallel lines form equal corresponding angles.

19. ∠2 > ∠1 and ∠2 > ∠A.

24. An equilateral triangle.

27. It is a right triangle.

28. They are complementary.

34. ∠2 and ∠3 are complementary.

36. Complements of the same angle are equal.

40. An exterior angle of a triangle is equal to the sum of the remote interior angles.

41. Substitution.

Chapter 6, Lesson 6 (pages 245–248)

1. ASA.

5. HL.

6. Parallel lines form equal corresponding angles.

9. Three.

12. Yes.

19. HL.

21. If two angles of a triangle are equal, the sides opposite them are equal.

24. In a plane, two lines perpendicular to a third line are parallel.

27. Substitution.

29. AAS.

32. AB = AC, AD = AE, and ∠A = ∠A.

40. Because ∠DAP = ∠EAP (corresponding parts of congruent triangles are equal).

42. SSS.

44. 45°.

50. Substitution.

Chapter 6, Review (pages 250–254)

1. Fold the figure along the proposed line of symmetry and see if the two halves coincide.

5. It is the perpendicular bisector of AB.

7. Equal corresponding angles mean that lines are parallel.

9. To the sum of the remote interior angles.

12. They are perpendicular.

14. Are equal, their sides are perpendicular.

16. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

26. Equilateral.

30. In a plane, two lines perpendicular to a third line are parallel.

33. In a plane, a line perpendicular to one of two parallel lines is also perpendicular to the other.

35. Concurrent.

39. ∠B = ∠CAD = 45°.

42. BA = CD by substitution.

48. ∠CDB = 2x°.

50. ∠ACB = 3x°.

Chapter 6, Algebra Review (page 256)

1. $\dfrac{7}{6}$.

2. $\dfrac{x}{20}$.

3. $\dfrac{x+10}{2x}$.

4. $\dfrac{4x-3}{x^3}$.

5. $\dfrac{2}{3}$.

6. $\dfrac{x^6}{8}$.

7. $\dfrac{5}{7}$.

8. $\dfrac{x+2}{2}$.

9. $\dfrac{3x}{4}$.

10. $\dfrac{1}{5}$.

11. $\dfrac{1}{6}$.

12. $\dfrac{2}{x}$.

13. $\dfrac{4x+2y-z}{8}$.

14. $\dfrac{4\pi r^3}{3}$.

15. $\dfrac{x-y}{y}$.

16. $\dfrac{2(x-3)}{2x-3}$ or $\dfrac{2x-6}{2x-3}$.

17. $\dfrac{x+2}{x^2-1}$.

18. $\dfrac{1+x}{x-y}$.

19. 1.

20. $\dfrac{x^3}{x-1}$.

Chapter 7, Lesson 1 (pages 260–264)

1. Two points determine a line.

3. Addition.

6. The sum of the angles of a quadrilateral is 360°.

9. A quadrilateral each of whose angles is a right angle is a rectangle.

10. Each angle of a rectangle is a right angle.

14. Lines that form right angles are perpendicular.

17. They seem to be parallel and equal.

21. They are equiangular (or each of their angles is a right angle).

26. $2x+2y=180$.

29. 144°.

31. SSS.

37. Triangles, quadrilaterals, and an octagon.

40. Six sides, three diagonals, and four triangles.

43. $n-3$.

46. 720°.

47. 120°.

50. $(n-2)180^\circ$.

52. Their sum is 360°.

54. They all appear to be smaller.

Chapter 7, Lesson 2 (pages 267–269)

2. Point.

3. See if the figure looks exactly the same when it is turned upside down.

4. Three.

5. Yes. The opposite sides of a parallelogram are equal.

8. An exterior angle of a triangle is greater than either remote interior angle.

11. The opposite sides of a parallelogram are parallel.

15. A line segment is bisected if it is divided into two equal segments.

17. The diagonals of a parallelogram bisect each other.

18. Two points are symmetric with respect to a point if it is the midpoint of the line segment connecting them.

p. 753

26. 10.

30. About 24°.

32. ∠AEB = $x^\circ$, ∠CED = $y^\circ$.

35. AD = 2DC.

37. 180.

45. All three.

Chapter 7, Lesson 3 (pages 272–275)

1. A quadrilateral is a parallelogram if its opposite sides are equal.

3. Parallel lines form equal corresponding angles.

4. In a plane, if a line is perpendicular to one of two parallel lines, it is also perpendicular to the other.

8. Supplementary interior angles on the same side of a transversal mean that lines are parallel.

13. SAS.

15. A quadrilateral is a parallelogram if its opposite sides are equal.

20. Equal alternate interior angles mean that lines are parallel.

21. A quadrilateral is a parallelogram if two opposite sides are both parallel and equal.

22. Line symmetry. The figure can be folded along a vertical line so that the two halves coincide.

26. No. Line symmetry (one vertical line).

29. They seem to be parallel.

32. The pairs of opposite angles.

37. A quadrilateral is a parallelogram if its diagonals bisect each other.

39. SSS.

43. A quadrilateral is a rectangle if all of its angles are equal.

47. 12.

53. They are the sides of the quadrilaterals above and below the braced square.

Chapter 7, Lesson 4 (pages 277–280)

1. 12.

2. Three.

5. 15.

7. All of the angles of a rectangle are equal.

10. A quadrilateral is a parallelogram if its opposite sides are equal.

12. All rectangles are parallelograms.

17. All of the sides of a rhombus are equal.

19. Two consecutive sides.

24. No. It must be a rectangle.

31. A quadrilateral is a parallelogram if its opposite sides are equal.

34. A quadrilateral all of whose angles are equal is a rectangle.

36. False.

39. True.

43. Its opposite sides are parallel.

Chapter 7, Lesson 5 (pages 282–285)

1. A trapezoid.

2. That it has exactly one pair of parallel sides.

4. That all of its angles are right angles (or that they are equal).

5. Yes. All rectangles are parallelograms.

8. Point.

10. They seem to be congruent.

11. Trapezoids.

13. No. The legs of a trapezoid must be equal for it to be isosceles.

16. The bases of a trapezoid are parallel.

17. Through a point not on a line, there is exactly one line parallel to the line.

19. The opposite sides of a parallelogram are equal.

22. If two sides of a triangle are equal, the angles opposite them are equal.

25. Parallel lines form supplementary interior angles with a transversal.

30. The base angles of an isosceles trapezoid are equal.

36. 75°.

40. Parallel lines form equal corresponding angles.

41. 105°.

46. AC and DB bisect each other.

47. A quadrilateral is a parallelogram if its diagonals bisect each other.

49. A trapezoid has only one pair of opposite sides parallel.

Chapter 7, Lesson 6 (pages 288–291)

2. A midsegment.

5. $DE\parallel AC$, and $DE=\dfrac{1}{2}AC$.

8. A midsegment of a triangle is parallel to the third side.

9. In a plane, two lines parallel to a third line are parallel to each other.

10. A midsegment of a triangle is half as long as the third side.

13. MNPQ is a parallelogram because two opposite sides are both parallel and equal.

17. Addition.

18. Substitution.

20. The sum of its diagonals.

24. ∠A = ∠C because the base angles of an isosceles trapezoid are equal.

28. (4, 7).

29. (8, 10).

32. 5.

35. They should be parallel.

38. They are congruent by SSS.

43. A rhombus.

49. MN seems to be parallel to AB and DC, the bases of the trapezoid.

Chapter 7, Review (pages 292–295)

2. Both pairs of its opposite sides are parallel.

4. AAS.

7. Substitution.

9. Four.

13. All of the sides and angles of a square are equal.

16. If a line divides an angle into two equal parts, it bisects the angle.

21. True.

27. True.

31. The sum of the angles of a quadrilateral is 360°.

33. The opposite angles of a parallelogram are equal.

36. D, E, and F.

42. GI ⊥ HM and ML ⊥ HM. In a plane, two lines perpendicular to a third line are parallel.

45. $2c$.

p. 754

49. A trapezoid.

51. No, because a trapezoid has two parallel sides.

56. IJKL is a trapezoid because $IJ\parallel LK$.

Chapter 7, Algebra Review (page 296)

1. 2.236.

2. 22.36.

3. 223.6.

4. 7.07.

5. 70.7.

6. $10\sqrt{5}$.

7. $100\sqrt{5}$.

8. $5\sqrt{2}$.

9. $50\sqrt{2}$.

10. $5\sqrt{x}$.

11. $x^2\sqrt{x}$.

12. $x\sqrt{\pi}$.

13. $2x^6\sqrt{3}$.

14. $\sqrt{49}\sqrt{3}+\sqrt{3}=7\sqrt{3}+\sqrt{3}=8\sqrt{3}$.

15. $\sqrt{150}=\sqrt{25}\sqrt{6}=5\sqrt{6}$.

16. $5\sqrt{7}$.

17. $5\sqrt{21}$.

18. $2\sqrt{15}$.

19. $6\sqrt{5}$.

20. 14.

21. 10.

22. $7\sqrt{6}$.

23. 11.

24. $24\sqrt{10}$.

25. 63.

26. $3\sqrt{5}$.

27. 10.

28. 23.

29. $9+\sqrt{3}$.

30. $4\sqrt{3}+1$.

31. $2\sqrt{x}+2\sqrt{y}$.

32. $x+2\sqrt{xy}+y$.

Chapter 8, Lesson 1 (pages 300–304)

1. A translation.

3. A rotation.

6. A reflection.

16. Exercises 9, 11, and 14.

17. Each has a vertical line of symmetry.

20. Yes.

25. A translation.

28. They are parallelograms because they have two sides that are both parallel and equal.

31. The opposite sides of a parallelogram are equal.

36. Betweenness of Rays Theorem.

37. Substitution.

43. A dilation.

44. They seem to be twice as long.

52. A translation.

56. A rotation.

Chapter 8, Lesson 2 (pages 308–311)

4. $2x$ units.

5. It is the segment’s perpendicular bisector.

7. A rotation.

9. 180°.

14. Its image looks the same.

18. B.

22. A reflection.

24. Yes, because it is the composite of two reflections in intersecting lines.

27. C and E.

28. 120°.

30. No. The figure does not look exactly the same upside down.

32. E.

35. E.

37. A translation.

41. A rotation.

44. If a point is reflected through a line, the line is the perpendicular bisector of the segment connecting the point and its image.

46. Its magnitude.

48. SAS.

Chapter 8, Lesson 3 (pages 314–316)

3. A reflection.

5. A glide reflection.

8. ∠PAB and ∠PCD.

10. The opposite sides of a parallelogram are equal.

14. A reflection.

17. The magnitude of the rotation.

18. The lines bisect these angles.

22. They are the perpendicular bisectors of these line segments.

25. Two figures are congruent if there is an isometry such that one figure is the image of the other.

27. D.

29. G.

35. No, because the two dogs are not mirror images of each other.

39. A translation and a reflection.

42. B′(14, 9), C′(9, 6).

43. $6\sqrt{2}$.

46. M(11, 4), N(12, 5), P(8, 1).

Chapter 8, Lesson 4 (pages 321–324)

1. Rotation (or point) symmetry.

6. It has reflection (line) symmetry with respect to a line through home plate and second base.

12. The point itself.

14. A line looks the same if it is rotated 180°.

16. A line has reflection symmetry because it can be reflected (folded) onto itself.

19. That rays OA and OD are opposite rays and that rays OB and OC are opposite rays.

22. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

25. They bisect each other.

30. Translation.

33. 22.5°.

34. No.

36. 16.

38. The y-axis.

42. Even.

45. Because it can be translated (in various directions) and look exactly the same.

Chapter 8, Review (pages 325–329)

6. No.

8. A glide reflection.

9. A rotation.

11. That it can be rotated so that it looks the same in three positions.

17. Yes.

20. A reflection.

24. Two figures are congruent if there is an isometry such that one figure is the image of the other.

29. A translation.

31. About 47°.

36. Because △ADC is the reflection image of △ABC.

37. It has reflection (line) symmetry with respect to line AC.

41. 45°.

47. No.

Chapter 9, Lesson 1 (pages 340–342)

3. Its area is equal to the sum of the areas of its nonoverlapping parts.

7. True, because △ADC ≅ △ABC.

9. True, because $a_{\triangle ADC}-a_{\triangle1}-a_{\triangle2}=a_{\triangle ABC}-a_{\triangle4}-a_{\triangle3}$.

14. $2x$ square units.

16. $(x+y)$ square units.

20. They are both concave octagons.

23. $3a+6b$.

26. They are congruent (SSS)

p. 755

and therefore equal in area.

30. No.

31. Yes. Each one contains two of the four triangles, all of whose areas are equal.

35. $\dfrac{1}{6}$ square unit.

37. 2 square units.

42. It is equilateral and equiangular.

43. BFED contains four of the congruent triangles, and ABCD contains two of them.

44. 4 square units.

49. They are both equal to AB.

50. Because DF = DC + CF = CF + FE = CE.

Chapter 9, Lesson 2 (pages 345–349)

1. 20 ft.

4. 3,600 in².

8. Four.

11. $a(b+c)$.

12. $ab$ and $ac$.

14. $(a+b)^2$.

19. Two congruent triangles.

22. $\sqrt{26}$.

25. No.

28. 2,592 nails.

30. 66 ft.

34. 6,400 square chains.

37. 84 cm².

38. 17.5%.

40. 21.6 ft long.

43. The total surface area of the walls.

47. 1,728 in².

48. 189 in².

Chapter 9, Lesson 3 (pages 353–357)

1. 6 square units.

5. $\dfrac{1}{4}c^2$.

6. The area of a right triangle is half the product of its legs.

7. The area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts (the Area Postulate).

9. Substitution.

15. They seem to be equal.

17. △ABC, 40 units; △DEF, 80 units.

21. They seem to stay the same.

27. $\alpha_{ABCD}\approx99.87$ square chains.

29. The area of a triangle is half the product of any base and corresponding altitude.

31. Substitution.

33. Division.

35. 84 square units.

36. 11.2 units.

41. In a plane, two points each equidistant from the endpoints of a line segment determine the perpendicular bisector of the line segment.

42. Betweenness of Points Theorem.

48. $\alpha_{\triangle AMB}=\alpha_{\triangle AMD}=20.25$ in²; $\alpha_{\triangle BMX}=\alpha_{\triangle DMY}=6.75$ in²; $\alpha_{\triangle CMX}=\alpha_{\triangle CMY}=13.5$ in².

52. About 1,200 shingles.

Chapter 9, Lesson 4 (pages 360–364)

1. That they are equal.

2. 2,800 ft².

6. Area = $\dfrac{1}{4}(s+s)(s+s)=\dfrac{1}{4}(2s)(2s)=s^2$.

8. 260 square units.

14. 1,369 square units.

17. A parallelogram. (Two of its opposite sides are both parallel and equal.)

18. 1 unit. $A=bh=(1)(1)=1$.

23. 30,794 ft².

26. 4.5 ft².

28. 8.25 ft².

32. 675 m².

35. 3,696 m².

41. Its area is $(a+b)h$.

43. In a plane, two lines perpendicular to a third line are parallel.

46. 165 square units.

50. △EBH, FHCG (and FHCD).

Chapter 9, Lesson 5 (pages 367–370)

2. 2, 2, and 4 units.

4. The Ruler Postulate.

5. The Protractor Postulate.

8. The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.

9. Substitution.

12. Substitution.

15. 12.

16. 1,225.

18. 37.

26. No. $256+225=481\neq484$.

33. 6 units.

35. 25 units.

37. $2ab$.

38. $(b-a)^2$.

43. The acute angles of a right triangle are complementary (so the sum of these angles is 90°).

45. A trapezoid.

49. It appears to bisect ABDE (that is, divide it into two congruent parts having equal areas).

Chapter 9, Review (pages 371–375)

1. 125 m².

3. 525.

4. Camp A, 68 paces; camp B, 64 paces; camp C, 60 paces.

8. 16 units.

12. The area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts.

14. ∠DEY.

16. AAS.

18. Corresponding parts of congruent triangles are equal.

20. Triangles with equal bases and equal altitudes have equal areas.

23. 11.

27. $4a$.

29. $4a+4b$.

30. $a(a+2b)$ or $a^2+2ab$.

34. 216 ft².

38. 332.5 ft².

40. It would be four times as great.

44. $\alpha_{\triangle AEF}=2x$ (because AF = 2AC and BH = EG).

46. $4x^2-xy$.

47. $4x-y$.

51. It would be rectangular.

Chapter 9, Algebra Review (page 376)

1. 500.

2. $\dfrac{1}{5}$ or 0.2.

3. 2.5.

4. ±10.

5. −3.

6. 42.

7. 15.