Notes — Chapter 4: Congruent Triangles

To the Student. Congruent triangles have the same size and shape. We begin our study of shapes with them. You will see how their shapes are determined by lengths of sides and sizes of angles. Shapes can also be described using coordinates, as you will see in the first lesson of this chapter. The distance and angle approach of congruence and the coordinate approach work together. Surveyors work out coordinates using distances and angles, and you will see that the Pythagorean Theorem lets you find distances from coordinates.

Key concepts in this chapter are: coordinates and the distance formula (use of the Pythagorean Theorem), and congruence and the ways to prove that triangles are congruent. Key techniques are: using coordinates to describe shapes and find distances, using and marking diagrams to show geometric relationships, and showing that triangles are congruent so that various segments and angles must be equal.

Lesson 1. Coordinates and Distances. Maps motivate the idea of a rectangular coordinate system and its coordinate grid. The Ruler Postulate gives us the numbers on the axes. The Pythagorean Theorem (Chapter 2, Lesson 6) gives us the distance formula. Examples in the exercises include the Geoboard, map reading, optics, plots of characteristics of birds and bats, architecture, and computer displays. The vector picture is introduced using water velocities.

Lesson 2. Polygons and Congruence. Congruence is the key idea here. Trace a polygon, if the tracing can be set down on another polygon so the two coincide then they are congruent. The lengths of corresponding segments and the sizes of corresponding angles must be equal. For triangles, the equality of corresponding parts defines congruence. Later, we will relate congruence to isometries. Examples used in the exercises include: a connect-the-dot figure, a patent drawing, an “art gallery” problem, and a fractal curve. There are many discovery exercises, and there is much practice with coordinates.

Lesson 3. ASA and SAS Congruence. A dramatic application of the ASA method from World War II opens the lesson. The lesson shows why the ASA and SAS Congruence Postulates should be true. The exercises show how useful these postulates are. Key terms such as included angles and opposite sides are defined. An exercise anticipates the names of types of triangles that will be formally given in Lesson 5. Applications include: binocular vision, measuring the Earth, and surveying.

Lesson 4. Congruence Proofs. A cartoon introduces a use of the ASA Postulate. A paragraph proof leads students to a second two-column proof. A discussion follows of the various ways proofs can be written. Applications are given to roof construction and estimating distances. There is practice bringing coordinate and congruence ideas together. There are exercises involving overlying triangles by folding paper that bring home the physical meaning of congruence.

Lesson 5. Isosceles and Equilateral Triangles. If two sides of a triangle are equal, the angles opposite them are equal. Jacobs presents Pappus’ beautiful proof of this result. He avoids “base angles” to let students think more easily of these triangles in various orientations. This lesson has a rich harvest of theorems and corollaries, and the exercises offer practice with proofs.

An application to the geometry of the hydrogen sulfide molecule is worth mention. This atom has a bond angle of 92.8°. The Pythagorean Theorem does not apply directly, but as an approximation it is good to within 2%. An Escher tiling is the Set III puzzler, where a mirror-image piece provides a final twist.

Lesson 6. SSS Congruence. Seeing that triangular braces make frameworks rigid motivates introduction of the powerful SSS congruence theorem. Jacobs gives Euclid’s proof.

Exercises include showing why the bisection constructions from Chapter 1 work.

A cautionary exercise shows that two triangles can have five pairs of parts equal and not be congruent. The geometry of linkages and of the plumb level is examined in the exercises. Vectors appear as forces that are in

balance if when arranged head to tail they form a triangle. Identifying types of triangles given in coordinate form gives practice with the use of coordinates and the distance formula.

Lesson 7. Constructions. Methods for copying segments, angles, and triangles complete our list of basic straightedge and compass constructions. Walk your compass down a line marking equal segments and you have the start of a ruler. Add in marks for halves, quarters, and so on by bisections, and you can pin down the coordinates of points as closely as you wish. Such constructions are background for the rulers, protractors, and coordinate systems of metric geometry — and they show how distances and shapes are determined by lengths and angles.

The exercises give a striking example of a construction discovered in 1743 in Sweden for determining where the frets go on a guitar, still a tricky problem. In making this construction, we use a ruler to shorten the work of making equal divisions on a base line. Other examples in the exercises include girders, jacks, and the reflection of light.