Lines & Angles • Topic 1 of 3

Angle Relationships

Two angles are complementary when they add to 90° and supplementary when they add to 180°. When two straight lines cross, they form two pairs of vertical angles, and vertical angles are always equal — this is the workhorse fact of the whole chapter. Any two angles that sit next to each other on a straight line form a linear pair and are supplementary. On the GRE the figure is not drawn to scale, so never assume an angle is 90° or that two angles are equal unless the problem states it, marks it with a square symbol, or you can prove it; instead, set the stated relationships equal and solve for the variable.

Two crossing lines forming two pairs of equal vertical anglesVertical & linear-pair anglesa°a°b°b°Vertical angles equal · a + b = 180°

✅ Solved examples

1. Two angles are complementary and one of them is 34°. Find the other.
Complementary means they sum to 90°. Other angle = 90° − 34° = 56°.
2. An angle is supplementary to 118°. Find the angle.
Supplementary angles sum to 180°. Angle = 180° − 118° = 62°.
3. Two vertical angles are given as (3x + 10)° and (5x − 30)°. Find the measure of each.
Vertical angles are equal: 3x + 10 = 5x − 30 → 40 = 2x → x = 20. Each angle = 3(20) + 10 = 70°.
4. Quantitative Comparison. An angle’s supplement is 4 times the angle itself. Quantity A: the measure of the angle. Quantity B: 40°. Which is greater?
Let the angle be a. Its supplement is 180 − a, and this equals 4a: 180 − a = 4a → 180 = 5a → a = 36°. Quantity A = 36°, Quantity B = 40°, so Quantity B is greater.

✏️ Practice — try these, take hints as needed

1. Find the complement of 27°.
Complements sum to 90°.
Subtract from 90.
90 − 27.
63°
2. Find the supplement of 95°.
Supplements sum to 180°.
Subtract from 180.
180 − 95.
85°
3. Two supplementary angles are in the ratio 2 : 3. Find the larger angle.
Let the parts be 2x and 3x.
2x + 3x = 180.
x = 36, larger = 3x.
108°
4. Two vertical angles measure (2x + 15)° and (x + 40)°. Find the measure of each angle.
Vertical angles are equal.
2x + 15 = x + 40.
x = 25, substitute back.
65°
5. An angle is equal to its own complement. Find the angle.
Let it be a; its complement is also a.
a + a = 90.
2a = 90.
45°

📝 Topic test — 8 questions

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