Chapter 7: A Tale of Three Intersecting Lines
When three lines meet pairwise (without all passing through a single point), they enclose a triangle. This humble shape , only three sides and three angles , is the cornerstone of geometry.
In this chapter, you will discover the angle sum property: the three angles of any triangle always add up to . You will see the exterior angle property, which connects an exterior angle to the two non-adjacent interior angles. You will learn the triangle inequality , the rule that decides whether three given lengths can actually be sides of a triangle.
These are not random rules; each one has a reason rooted in the basic facts about lines and angles you learned in Chapter 5. By the end of this chapter, you will be able to find any missing angle or side of a triangle given a few clues , and explain why.
Triangles are everywhere: in roof trusses, in bridge supports, in surveying, in computer graphics. The rules you learn here are the same ones engineers use today.
What's inside
- Types of triangles , by sides (scalene, isosceles, equilateral) and by angles.
- Angle sum property , why the three angles always add to .
- Exterior angle property , the bonus angle outside the triangle.
- Triangle inequality , when three lengths can form a triangle.
- Equilateral and isosceles tricks , using symmetry to find angles fast.
Key results
| Property | Statement |
|---|---|
| Angle sum | |
| Exterior angle | Exterior angle sum of two opposite interior angles |
| Triangle inequality | The sum of any two sides third side |
| Equilateral | All sides equal all angles |
| Isosceles | Two sides equal angles opposite those sides are equal |
| Right triangle | One angle ; other two sum to |
How to read this chapter
Cut three paper triangles. Tear off the three corners of each and place them next to each other at a single point , you will see they form a straight line (). This is the most striking visual proof of the angle sum property.