Lines & Angles • Topic 3 of 3

Angle Pairs

Four relationships clear up most plain-figure CAT questions. Complementary angles sum to 90°; supplementary angles sum to 180°. A linear pair is two adjacent angles whose non-common arms form a straight line, so they are supplementary and add to 180°. When two straight lines intersect, the angles directly across from each other — vertically opposite angles — are equal, and each adjacent pair forms a linear pair. Around a single point the angles total 360°. The fast technique is to assign one unknown and chain these rules: if a question gives "one angle is twice its complement", write x + 2x... no — write the angle and its complement as x and 90° − x and translate the words into one equation. Watch the wording trap: "complement" lives in the 90° world, "supplement" in the 180° world, and angle bisectors of a linear pair are always perpendicular, a result CAT slips into harder diagrams.

✅ Solved examples

1. An angle is 34°. Find its complement and its supplement.
Complement = 90° − 34° = 56°. Supplement = 180° − 34° = 146°.
2. Two angles form a linear pair and are in the ratio 4 : 5. Find both angles.
They sum to 180°: 4k + 5k = 180 ⇒ k = 20 ⇒ angles are 80° and 100°.
3. An angle is 20° more than its complement. Find the angle.
x + (x − 20) = 90 ⇒ 2x = 110 ⇒ x = 55°.
4. Two straight lines intersect; one of the four angles is 70°. Find all four angles.
Vertically opposite to it is also 70°; the other pair are linear-pair supplements: 110° each. So 70°, 110°, 70°, 110°.

✏️ Practice — try these, take hints as needed

1. Find the supplement of an angle whose complement is 28°.
Find the angle first: 90 − 28.
Angle = 62°.
Supplement = 180 − 62.
118°
2. Two supplementary angles are in the ratio 7 : 5. Find the larger angle.
They sum to 180°.
7k + 5k = 180.
k = 15, larger = 7k.
105°
3. An angle is three times its supplement. Find the angle.
Let angle = x, supplement = 180 − x.
x = 3(180 − x).
4x = 540.
135°
4. Two lines intersect so that one angle is 5 times another adjacent angle. Find the smaller angle.
Adjacent angles form a linear pair.
x + 5x = 180.
6x = 180.
30°
5. An angle is equal to its own complement. Find the angle.
x = 90 − x.
2x = 90.
Solve.
45°

📝 Topic test — 8 questions

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