Angles & Angle Relationships

Every time two rays share a starting point, they form an angle. Angles are measured in degrees (the symbol is a tiny circle: °). A full rotation around a point is 360°. Understanding angle types and their relationships lets you find unknown measurements without a protractor—just by reasoning.

Classifying Angles

Type	Measure	Picture Cue
Acute	Between 0° and 90°	Looks "sharp" — narrower than a corner of a page
Right	Exactly 90°	A perfect square corner; marked with a small square
Obtuse	Between 90° and 180°	Wider than a page corner but not a flat line
Straight	Exactly 180°	A flat line—the two rays point in opposite directions

Complementary & Supplementary Angles

Two angles are complementary when their measures add to 90°. Two angles are supplementary when their measures add to 180°. They do not need to be next to each other—only their measures matter.

Complementary: angle A + angle B = 90° Supplementary: angle A + angle B = 180°

Memory Trick

Complementary → Corner (90°). Supplementary → Straight line (180°).

Vertical & Adjacent Angles

When two lines cross, they create two special relationships:

Adjacent angles share a side and a vertex. They sit next to each other and are supplementary (they form a straight line together).
Vertical angles are across from each other at the intersection. They are always equal.

Picture two lines crossing like an X. Label the four angles 1, 2, 3, 4 going clockwise. Angles 1 and 3 are vertical (equal). Angles 2 and 4 are vertical (equal). Angles 1 and 2 are adjacent (supplementary).

Example 1 — Complementary Angles

One angle measures 37°. Find its complement.

Complementary angles sum to 90°.

Unknown = 90° − 37° = 53°.

Example 2 — Vertical Angles & Supplements

Two lines intersect. One of the four angles is 125°. Find the other three.

The vertical angle is also 125°.

Each adjacent angle is supplementary: 180° − 125° = 55°.

The four angles are 125°, 55°, 125°, 55°.

Example 3 — Algebra with Angle Relationships

Two supplementary angles measure (3x + 10)° and (2x)°. Find x and both angles.

Set up: (3x + 10) + 2x = 180.

Combine: 5x + 10 = 180.

Solve: 5x = 170, so x = 34.

Angles: 3(34) + 10 = 112° and 2(34) = 68°.

Check: 112 + 68 = 180. Correct.

Common Mistake

Students sometimes mix up complementary (90°) and supplementary (180°). Always double-check which sum the problem asks for before subtracting.

Practice Problems

1. Classify each angle: (a) 17° (b) 90° (c) 143° (d) 180°

Show Solution

(a) Acute (b) Right (c) Obtuse (d) Straight

2. Find the complement of 58°.

Show Solution

90° − 58° = 32°

3. Find the supplement of 47°.

Show Solution

180° − 47° = 133°

4. Two lines cross. One angle is 72°. Find the other three angles.

Show Solution

Vertical angle = 72°. Each adjacent angle = 180° − 72° = 108°. The four angles are 72°, 108°, 72°, 108°.

5. Two complementary angles are (4x)° and (x + 15)°. Find both angle measures.

Show Solution

4x + x + 15 = 90 → 5x = 75 → x = 15. Angles: 4(15) = 60° and 15 + 15 = 30°. Check: 60 + 30 = 90.

Lesson Summary

Angles are classified as acute (<90°), right (=90°), obtuse (90°–180°), or straight (=180°).
Complementary angles sum to 90°; supplementary angles sum to 180°.
Vertical angles are equal. Adjacent angles on a line are supplementary.
You can use these relationships to set up equations and solve for unknowns.