♥ Support Post Sign in Join
Learn · Mathematics · Grade 9 · Triangles

Angles in a triangle and the triangle inequality

Interior and exterior angles, the angle sum, isosceles and equilateral triangles, and which three lengths can make a triangle.

High school Lesson 6 of 31 Interactive figure Practice

First, the rules

180° and the exterior angle

★ Angle sum
Examples 1, 2
A + B + C = 180°
Two angles known, the third is free.
★ Exterior angle
Example 1
The exterior angle at a vertex equals the sum of the two remote interior angles (and 180° minus the interior angle at that vertex).
★ Isosceles and equilateral
Example 2
Two equal sides ⇒ the base angles are equal. Three equal sides ⇒ three 60° angles.
Bigger side, bigger angle
Example 2
The longest side faces the largest angle; equal sides face equal angles.
★ Triangle inequality
Example 3
|b − c| < a < b + c
Each side is shorter than the other two together and longer than their difference; otherwise the triangle does not close.
Right triangle
A = 90° ⇒ B + C = 90° (the acute angles are complementary).

See it

Drag a corner — the sum never moves

Pull any vertex. The three interior angles change, their total stays 180°, and the orange exterior angle at C always equals A + B.

Step by step

Worked examples

1

Chasing angles

angle sum · exterior angle

In triangle ABC, A = 48° and B = 67°. Find C and the exterior angle at C.

48°67°?extABC

Ask yourself: The third angle comes from 180°; the exterior angle is what is left on the straight line — or simply A + B.

Show the solution
  1. C = 180° − 48° − 67° = 65°.
  2. Exterior angle at C: 180° − 65° = 115°, and indeed 48° + 67° = 115°.

Answer C = 65°, exterior 115°

2

An isosceles triangle

equal sides, equal angles

In triangle ABC, |AB| = |AC| and the angle at A is 40°. Find the other angles. Which side is the longest?

Ask yourself: Equal sides AB and AC face which angles? Those two are equal.

Show the solution
  1. |AB| = |AC| means the angles facing them are equal: B = C.
  2. 40° + 2B = 180° ⇒ B = C = 70°.
  3. The largest angle (70°) faces the longest side: BC, the base, is longer than the two equal legs. (The smallest angle, 40°, faces the shortest sides — both legs.)

Answer B = C = 70°; the base BC is the longest side

3

Can these sides make a triangle?

triangle inequality

(a) Can 4, 7 and 12 be the sides of a triangle? (b) Two sides of a triangle are 5 and 9. Between which values must the third side lie? If the third side is a whole number, how many triangles are possible?

Ask yourself: Check the two shortest against the longest: is their sum bigger?

Show the solution
  1. (a) 4 + 7 = 11 < 12: the two short sides cannot reach each other across 12. No triangle.
  2. (b) 9 − 5 < x < 9 + 5, i.e. 4 < x < 14.
  3. Whole numbers strictly between 4 and 14: 5, 6, …, 13 — nine values, nine triangles.

Answer (a) no · (b) 4 < x < 14, nine whole-number triangles

Practice: angles and sides

0 / 0

Find a missing angle, an exterior angle, the base angles of an isosceles triangle, or the range for a third side. Your best score is kept in this browser only.

About this lesson

Angles in a triangle: the interior angles add up to 180°, an exterior angle equals the sum of the two remote interior angles, isosceles and equilateral triangles, the biggest side faces the biggest angle, and the triangle inequality — which three lengths can make a triangle. Drag a triangle and watch the angles, then practise.

Part of Learn mathematics · Grade 9 · Triangles. Check a number with the scientific calculator or draw a function with the function plotter. Everything on this page runs in your browser; nothing you type is sent or stored, and your practice score stays in this browser only. Spotted an error? Tell us.