Lesson 1 — Quadrilaterals
Big idea: A quadrilateral is a four-sided polygon. Its angles add to 360°.
Words to know
| Word | Meaning |
|---|---|
| Convex polygon | No side, when extended, goes inside the polygon. |
| Concave polygon | At least one side, when extended, goes inside the polygon. |
| Diagonal | A segment joining two nonconsecutive vertices of a polygon. |
| Rectangle | A quadrilateral with four right angles. |
The theorem
Theorem 24: The sum of the angles of a quadrilateral is 360°.
Corollary: A quadrilateral is equiangular iff it is a rectangle.
Concepts to know
- A quadrilateral has 4 sides, 4 vertices, and 2 diagonals.
- Draw one diagonal to split a quadrilateral into two triangles.
- Each triangle has an angle sum of 180°, so the quadrilateral has 2 × 180° = 360°.
- A convex quadrilateral has diagonals inside; a concave one has a diagonal outside.
Homework helper
| If the homework asks… | Do this… |
|---|---|
| Find a missing angle | Subtract the known angles from 360°. |
| Classify as convex or concave | Check whether a side extended enters the figure. |
| Draw diagonals | Connect nonconsecutive vertices. |
| Prove a quadrilateral is a rectangle | Show all four angles are 90°. |
| Use the angle sum | Split into two triangles if you need a reason. |
Worked example
A quadrilateral has angles 80°, 100°, and 120°. Find the fourth.
80 + 100 + 120 = 300
360 − 300 = 60°The fourth angle is 60°.
Watch out
- The sum is 360°, not 180°.
- A diagonal joins nonconsecutive vertices, not adjacent ones.
- A concave quadrilateral has a “dent.”
- An equiangular quadrilateral is a rectangle; all four angles are 90°.