Triangle Basics

Classify by sides and angles, and use the 180° angle sum.

The idea

A triangle is the figure formed by three points not on one line, joined in pairs. In every triangle, the three interior angles add to $180°$. A short construction with a parallel line proves this, and most of what follows about triangles rests on it.

Two consequences follow at once. Knowing two angles determines the third. And at most one angle can be $90°$ or more, since two such angles would already use up the $180°$.

The second consequence is why one angle can classify a triangle: a triangle is acute when every angle is under $90°$, right when one angle is exactly $90°$, and obtuse when one angle is over $90°$, and exactly one of the three names applies. A separate classification counts equal sides: scalene, isosceles and equilateral for none, two, or all three sides equal. The two classifications are independent, so a full description gives one name from each, as in right isosceles.

Extending one side past a vertex produces an exterior angle, which equals the sum of the two interior angles not adjacent to it.

Ways to work on it

• Walkthrough. Classify by sides and angles; use the 180° sum and exterior angles.
• Proof. See why the angles sum to 180° — the parallel-line picture.
• Practice. Recover the missing angle from the other two.
• Hardest. Angles given by a ratio or by algebra — find the largest.

Not sure where to start? Take the ten-question placement test.