Angle Relationships Examples in Math

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

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

Concept Recap

Fundamental relationships between pairs of angles: supplementary angles sum to 180°180°, complementary angles sum to 90°90°, vertical angles are equal, and adjacent angles share a common ray.

Think of opening a book flat on a table—the two pages form supplementary angles (they add to a straight line, 180°180°). Now think of the corner of a room where two walls meet the floor—those two angles are complementary (they add to a right angle, 90°90°). When two lines cross like an X, the opposite angles are always equal—those are vertical angles.

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: Pairs of angles formed by intersecting or adjacent rays add to 180 degrees, 90 degrees, or are equal.

Common stuck point: The procedure for angle relationships is the easy part; the trap is confusing supplementary (180°180°) with complementary (90°90°). Asking "Do these two angles share a vertex or a straight line so their measures are forced to add to 180, add to 90, or be equal?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

Sense of Study hint: Ask: Do these two angles share a vertex or a straight line so their measures are forced to add to 180, add to 90, or be equal?

Worked Examples

Example 1

easy
Two angles are supplementary. One angle measures 73°73°. Find the other angle.

Answer

The other angle is 107°107°.

First step

1
Step 1: Supplementary angles sum to 180°180°.

Full solution

  1. 2
    Step 2: Let the other angle be xx. Then 73°+x=180°73° + x = 180°.
  2. 3
    Step 3: x=180°−73°=107°x = 180° - 73° = 107°.
Supplementary angles are two angles whose measures add to 180°. They don't need to be adjacent — any two angles summing to 180° are supplementary. A straight line is formed when supplementary angles are adjacent, which is one common way they appear in geometry problems.

Example 2

medium
Two lines intersect forming four angles. One angle is 124°124°. Find all four angles.

Example 3

medium
∠A\angle A and ∠B\angle B are supplementary. ∠A=3x+10°\angle A = 3x + 10° and ∠B=x+30°\angle B = x + 30°. Find each angle.

Example 4

medium
Two angles form a linear pair. The supplement is four times the angle. Find both.

Example 5

medium
∠A=(2x+5)°\angle A = (2x + 5)° and ∠B=(3x−10)°\angle B = (3x - 10)° are vertical angles. Find xx and each angle.

Practice Problems

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

Example 1

easy
Angle AA and angle BB are complementary. If ∠A=4x+3°\angle A = 4x + 3° and ∠B=2x+9°\angle B = 2x + 9°, find both angles.

Example 2

hard
Three angles share a common vertex and together form a straight line. The angles are in the ratio 1:2:31:2:3. Find each angle.

Example 3

easy
Two lines intersect. One angle is 25°25°. Find the other three angles formed.

Example 4

medium
Two angles are complementary. The larger is 14°14° more than the smaller. Find both angles.

Example 5

medium
Two vertical angles measure (5x−12)°(5x - 12)° and (3x+8)°(3x + 8)°. Find xx and each angle.

Example 6

easy
An angle measures 34°34°. Find its complement and supplement.

Example 7

hard
An angle's supplement is 30°30° less than three times its complement. Find the angle.

Example 8

medium
Two intersecting lines form four angles. Three of them are aa, bb, and cc with bb vertical to aa and cc adjacent to aa. If a=78°a = 78°, find bb and cc.

Example 9

medium
Two angles are supplementary, and their measures are in the ratio 7:57:5. Find both angles.

Example 10

hard
Four rays emanate from a point, forming four consecutive angles that sum to 360°360°. The angles are xx, 2x2x, 3x3x, and 4x4x. Find each angle.

Example 11

easy
Two angles are supplementary and one is a right angle. Find the other.

Example 12

hard
Three rays from a single point form angles ∠1,∠2,∠3\angle 1, \angle 2, \angle 3 around the point that sum to 360°360°. ∠1=90°\angle 1 = 90° and ∠2=2∠3\angle 2 = 2\angle 3. Find ∠2\angle 2 and ∠3\angle 3.

Example 13

hard
∠A\angle A and ∠B\angle B are supplementary. If ∠A\angle A is half of ∠B\angle B, find each angle.

Example 14

easy
Three angles meet at a point and are 120°,100°120°, 100°, and x°x°. Find xx.

Example 15

challenge
Two adjacent angles form a straight line. The angle bisectors of each form a new angle between them. Prove that this new angle is always 90°90°.

Example 16

medium
Two angles are supplementary and the difference of their measures is 40°40°. Find both angles.

Example 17

hard
In a figure, ray OCOC bisects ∠AOB\angle AOB. If ∠AOC=(3x+4)°\angle AOC = (3x + 4)° and ∠COB=(5x−12)°\angle COB = (5x - 12)°, find xx and ∠AOB\angle AOB.
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Background Knowledge

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

angles