DWG NO. 01.4 — Lesson 4 of 5

Angle Pair Relationships

Unit 1: Foundations of Geometry · ~15–30 min

Objective

Identify complementary, supplementary, adjacent, and vertical angle pairs, and use their relationships to solve for unknown angle measures.

Four relationships worth knowing

Pair	Definition
Adjacent	Two angles that share a vertex and a side, with no overlapping interior.
Complementary	Two angles whose measures add up to 90°.
Supplementary	Two angles whose measures add up to 180°.
Linear pair	Adjacent angles whose non–shared sides form a straight line (opposite rays) — always supplementary.
Vertical angles	The pair of non–adjacent angles formed opposite each other when two lines cross — always congruent.

The last two are theorems worth remembering by name: the Linear Pair Postulate (a linear pair is always supplementary) and the Vertical Angles Theorem (vertical angles are always congruent).

Two intersecting lines form four angles. ∠1 & ∠3 and ∠2 & ∠4 are vertical pairs; any two angles next to each other (like ∠1 & ∠2) form a linear pair.

Worked Example 1 · Linear pair

Problem∠1 and ∠2 form a linear pair. m∠1 = (2x+14)° and m∠2 = (3x+6)°. Find x and both measures.

1.A linear pair is always supplementary: (2x+14) + (3x+6) = 180

2.5x + 20 = 180 → 5x = 160 → x = 32

3.m∠1 = 2(32)+14 = 78°. m∠2 = 3(32)+6 = 102°. Check: 78 + 102 = 180 ✓

x = 32 • m∠1 = 78° • m∠2 = 102°

Worked Example 2 · Complementary angles

ProblemTwo angles are complementary. One measures twice the other. Find both measures.

1.Let the smaller angle be x, so the larger is 2x. x + 2x = 90

2.3x = 90 → x = 30

The two angles measure 30° and 60°.

Worked Example 3 · Vertical angles

ProblemTwo lines intersect. One of the four angles formed measures 47°. Find the other three.

1.Its vertical angle (directly opposite) is congruent to it: 47°.

2.Its two neighbors are linear–pair partners: 180° − 47° = 133° each.

The four angles measure 47°, 133°, 47°, 133° going around the intersection.

Guided practice

1.Two angles are supplementary. One is 3 times the other. Find both measures.
Show answer
x + 3x = 180 → 4x = 180 → x = 45. The angles measure 45° and 135°.
2.∠3 and ∠4 form a linear pair. m∠3 = (5x−8)°, m∠4 = (3x+20)°. Find x and each measure.
Show answer
5x−8+3x+20 = 180 → 8x+12 = 180 → 8x = 168 → x = 21. m∠3 = 97°, m∠4 = 83°.
3.∠1 and ∠2 are vertical angles. m∠1 = (4x+10)°, m∠2 = (6x−30)°. Find x and the measure of each angle.
Show answer
Vertical angles are congruent: 4x+10 = 6x−30 → 40 = 2x → x = 20. Both angles measure 4(20)+10 = 90°.
4.∠A and ∠B are complementary. m∠A = 22°. Find m∠B.
Show answer
m∠B = 90° − 22° = 68°
5.Two lines intersect, forming four angles. One measures 90°. What must be true about the other three, and why?
Show answer
All three also measure 90°. Its linear–pair partners must each be 180° − 90° = 90°, and its vertical angle is congruent to it by the Vertical Angles Theorem. (This is exactly what happens when two lines are perpendicular.)

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