Triangle Angle Sum Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Triangle Angle Sum.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The three interior angles of any triangle always sum to exactly 180°180°, so knowing two angles determines the third.

Tear off the three corners of any paper triangle and line them up—they always form a straight line (180°180°). No matter how pointy or flat the triangle is, the angles always add up the same way, like three puzzle pieces that always complete a half-turn.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: The interior angles of any triangle add to exactly 180°180°, so two of them fix the third.

Common stuck point: The procedure for triangle angle sum is the easy part; the trap is using 360°360° for a triangle. Asking "Am I finding a missing interior angle of one triangle from the others, using a total of 180°180°?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Am I finding a missing interior angle of one triangle from the others, using a total of 180°180°?

Worked Examples

Example 1

easy
A triangle has angles 55°55° and 72°72°. Find the third angle.

Answer

The third angle is 53°53°.

First step

1
Step 1: Recall the Triangle Angle Sum Theorem: the three interior angles of any triangle sum to 180°180°.

Full solution

  1. 2
    Step 2: Let the third angle be xx. Then 55°+72°+x=180°55° + 72° + x = 180°.
  2. 3
    Step 3: Solve: x=180°−55°−72°=180°−127°=53°x = 180° - 55° - 72° = 180° - 127° = 53°.
The Triangle Angle Sum Theorem states that the interior angles of every triangle add to exactly 180°. This holds for all triangles — acute, right, and obtuse. To find a missing angle, subtract the sum of the known angles from 180°.

Example 2

medium
In an isosceles triangle, the vertex angle is 40°40°. Find the two base angles.

Example 3

medium
The angles of a triangle are xx, x+20°x+20°, and x+40°x+40°. Find each angle.

Example 4

medium
A triangle has angles (3x+5)°(3x + 5)°, (2x−10)°(2x - 10)°, and (x+5)°(x + 5)°. Find each angle.

Example 5

medium
In △ABC\triangle ABC, ∠A:∠B:∠C=3:4:5\angle A : \angle B : \angle C = 3 : 4 : 5. Classify the triangle.

Example 6

medium
△ABC\triangle ABC is isosceles with AB=ACAB = AC. The vertex angle ∠A=96°\angle A = 96°. Find each base angle.

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Can a triangle have angles of 90°90°, 91°91°, and 1°1°? Justify your answer.

Example 2

hard
In △ABC\triangle ABC, ∠A=2x+10°\angle A = 2x + 10°, ∠B=3x−5°\angle B = 3x - 5°, ∠C=x+15°\angle C = x + 15°. Find all three angles.

Example 3

easy
A triangle has angles 42°42° and 88°88°. Find the third angle.

Example 4

easy
Two angles of a triangle measure 30°30° and 30°30°. Find the third angle and name the triangle by sides.

Example 5

medium
In an isosceles triangle, each base angle is 5°5° more than twice the vertex angle. Find all three angles.

Example 6

medium
The angles of a triangle are in the ratio 2:3:52:3:5. Find the largest angle and classify the triangle.

Example 7

medium
In △ABC\triangle ABC, ∠A\angle A is twice ∠B\angle B, and ∠C\angle C is 30°30° more than ∠B\angle B. Find all three angles.

Example 8

easy
Can a triangle have angles 80°80°, 80°80°, and 20°20°? If so, classify it.

Example 9

medium
A triangle has angles xx, 2x2x, and 3x3x. What kind of triangle is it?

Example 10

hard
In △ABC\triangle ABC, ∠A\angle A exceeds ∠B\angle B by 12°12°, and ∠C\angle C is one-third of ∠B\angle B. Find each angle.

Example 11

hard
In △ABC\triangle ABC, ∠A=2∠B\angle A = 2 \angle B and ∠C=3∠B−20°\angle C = 3 \angle B - 20°. Find all three angles.

Example 12

easy
Can a triangle have angles 90°90°, 45°45°, and 45°45°? If so, what is it called?

Example 13

medium
Two angles of a triangle are equal and the third is 48°48° more than each. Find each angle.

Example 14

hard
In △ABC\triangle ABC, the angle bisector from AA meets BCBC at DD. If ∠B=70°\angle B = 70° and ∠C=50°\angle C = 50°, find ∠ADB\angle ADB.

Example 15

easy
Is a triangle with angles 100°100°, 40°40°, and 40°40° valid? If so, classify it.

Example 16

hard
In △ABC\triangle ABC, the angle at AA is twice the angle at BB minus the angle at CC. If ∠B=50°\angle B = 50° and ∠C=40°\angle C = 40°, is the data consistent? Find ∠A\angle A.

Example 17

medium
An exterior angle of a triangle is 110°110°. One remote interior angle is twice the other. Find the two remote interior angles.

Example 18

challenge
In △ABC\triangle ABC, the angle bisectors of ∠B\angle B and ∠C\angle C meet at II. Show that ∠BIC=90°+12∠A\angle BIC = 90° + \tfrac{1}{2}\angle A. Verify it when ∠A=50°\angle A = 50°.

Example 19

hard
In △ABC\triangle ABC, ∠A\angle A is 30°30° larger than ∠B\angle B, and ∠C\angle C equals ∠B\angle B. Find each angle.
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Background Knowledge

These ideas may be useful before you work through the harder examples.

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