Complementary, supplementary, vertical, linear — four names, two numbers.
The problem
Two angles are complementary. One is 18° more than the other. Find both.
Complementary means the two add to 90°. So x + (x + 18) = 90, giving 2x = 72 and x = 36. The angles are 36° and 54°. A and B use 180° — that is supplementary, the other word.
add to 90°
C comes first — 90 is smaller
add to 180°
S comes second — 180 is bigger
C before S in the alphabet, 90 before 180 on the number line. That is the whole memory trick.
They do not have to touch. Two angles in different corners of a diagram can still be complementary — the definition is only about the sum.
“18° more than the other” tells you which is which.
smaller = x
start from the smaller
larger = x + 18
18 more
x + (x + 18) = 90
complementary
2x = 72, x = 36
− 18, then ÷ 2
x = 36 answers only half the question. The 54 needs the extra line, and it is the line that most often gets skipped.
Two crossing lines make four angles and only two sizes.
across the X — equal
side by side — 180°
Angles that look next to each other are supplementary, not equal. Only the pair directly opposite through the crossing point are equal.
(3x + 10) and (2x − 5) are supplementary
so they total 180
5x + 5 = 180
collect
x = 35
− 5, ÷ 5
angles = 115° and 65°
substitute back
115 + 65 = 180 ✓ — always confirm the sum.
If the two expressions were vertical angles instead, you would set them equal to each other rather than to 180.
| Pair | Where | Rule |
|---|---|---|
| complementary | anywhere | sum 90° |
| supplementary | anywhere | sum 180° |
| linear pair | adjacent, on a straight line | sum 180° |
| vertical | opposite, at a crossing | equal |
| adjacent | share a side and vertex | no rule by itself |
| angles on a straight line | three or more, side by side on one line | sum 180° |
| angles round a point | all the way round | sum 360° |
“Adjacent” on its own tells you nothing numerical — it only describes position.
Vertical angles measure (4x − 12)° and (2x + 30)°. Find each angle.
4x − 12 = 2x + 30
vertical: set them equal
2x = 42, x = 21
− 2x, + 12, ÷ 2
4(21) − 12 = 72°, 2(21) + 30 = 72°
both sides agree
Different: there is no 90 or 180 in the equation at all — the two expressions are equal.
Writing (4x − 12) + (2x + 30) = 180 gives x = 27 and angles 96° and 84°. They are not equal, so they cannot be vertical angles — the substitution exposes the slip.
Two angles are supplementary. One is 5° less than three-halves of the other. Find both.
x + ((32)x − 5) = 180
supplementary
2x + 3x − 10 = 360
× 2 clears the fraction
5x = 370, x = 74°
+ 10, ÷ 5
(32)(74) − 5 = 111 − 5 = 106°
74 + 106 = 180 ✓
Different: multiply every term by 2, the 5 and the 180 included, before collecting.
Find the complement and the supplement of 124°.
90 − 124 = −34
negative: no complement
180 − 124 = 56°
the supplement exists
Different: only acute angles have complements; an obtuse angle has a supplement only — and that supplement is acute, so two obtuse angles can never be supplementary (their sum is already over 180°).
Two angles are complementary and their measures are in the ratio 1 : 5. Find both.
k + 5k = 90
a ratio 1 : 5 means k and 5k
6k = 90, k = 15
÷ 6
15° and 75°
k is the multiplier, not an angle — multiply back
Two lines cross. A 70° angle is vertical to (x + 2y)° and forms a linear pair with (2x − y)°. Find x and y.
x + 2y = 70
vertical: equal
2x − y = 110
linear pair: 180 − 70
x + 2(2x − 110) = 70 → 5x = 290
y = 2x − 110, substitute
x = 58, y = 6
58 + 12 = 70 ✓, 116 − 6 = 110 ✓
Different: two unknowns need two relations — one from each pair the figure shows.
Call the angle A. Its complement is 90 − A and its supplement is 180 − A — then translate the sentence.
supplement = 3 × complement
the sentence
180 − A = 3(90 − A)
write each one in terms of A
180 − A = 270 − 3A
expand
2A = 90, A = 45
+ 3A, − 180, ÷ 2
Check: supplement 135°, complement 45°, and 135 = 3 × 45 ✓
Chained wording works from the inside out. “The supplement of the complement of 40°”: the complement of 40° is 50°, and the supplement of 50° is 130°.
Once A is found, many questions ask for its supplement or the larger of two angles. Re-read the last line before you choose — x on its own is the commonest wrong pick.
UPCAT-type items rarely ask for a plain complement. They describe an angle through its complement or supplement, or put two unknowns in a crossing-lines figure. Write one equation per relation — vertical angles give “equal”, a linear pair gives “180” — and two relations solve two unknowns. Ten-second check: supplement minus complement is always 90°.
The problem
Two supplementary angles are in the ratio 4 : 5. Find both.
Write them as 4k and 5k. Supplementary means they total 180°, so 9k = 180 and k = 20. That gives 80° and 100°. A is the ratio halved — it sums to 90, not 180.
Every lesson in the UPCAT Math Review has one of these. 30 questions, marked as you answer, with the working shown for each.
Question 1 of 30
What is the supplement of the complement of 62°?
Complement of 62° is 90 − 62 = 28°; the supplement of 28° is 180 − 28 = 152°. (118° is the supplement of 62° itself.)
Question 2 of 30
Two vertical angles measure (3x + 10)° and (5x − 30)°. What is the value of x?
Vertical angles are equal: 3x + 10 = 5x − 30 → 40 = 2x → x = 20. Treating them as supplementary (8x − 20 = 180) gives 25; 70 is the angle itself, 3(20) + 10.
Question 3 of 30
In the figure, the two angles form a linear pair and one is 4 times the other. Find the smaller angle.
A linear pair adds to 180°: x + 4x = 180 → x = 36°. The larger is 144°. (18° comes from using 90°.)
Question 4 of 30
In the figure, two lines intersect. Find the measure of ∠1.
The two labelled angles are vertical angles, so 5x − 20 = 3x + 30 → x = 25 and each measures 105°. ∠1 is adjacent to them, forming a linear pair: 180 − 105 = 75°. (105° answers the wrong angle.)
Question 5 of 30
Two angles are complementary. One of them is 12° less than twice the other. Find the supplement of the larger angle.
Let the smaller be x: x + (2x − 12) = 90 → 3x = 102 → x = 34, so the angles are 34° and 56°. The supplement of the larger is 180 − 56 = 124°. (146° is the supplement of the smaller.)
Question 6 of 30
An angle measures 30° more than twice its supplement. Find the angle.
Let the angle be A; its supplement is 180 − A. A = 2(180 − A) + 30 → 3A = 390 → A = 130°. Using the complement, 90 − A, gives 70°; 50° is the supplement, not the angle asked for.
Question 7 of 30
Two angles are supplementary and their measures are in the ratio 2 : 7. What is the complement of the smaller angle?
2x + 7x = 180 → x = 20, so the angles are 40° and 140°. The complement of 40° is 90° − 40° = 50°. 40° is the smaller angle itself; 20° shares 90° instead of 180° in the ratio.
Question 8 of 30
The supplement of an angle is 4 times its complement. Find the angle.
180 − A = 4(90 − A) → 180 − A = 360 − 4A → 3A = 180 → A = 60°. Check: supplement 120°, complement 30°, and 120 = 4 × 30.
Question 9 of 30
∠A and ∠B are complementary, and ∠B and ∠C are supplementary. If ∠C = 4∠A, find ∠B.
∠B = 90 − A, so ∠C = 180 − (90 − A) = 90 + A. Then 90 + A = 4A → A = 30°, ∠B = 60° and ∠C = 120°. 30° is ∠A; treating ∠A and ∠C as supplementary (5A = 180) gives 36°.
Question 10 of 30
Which of the following statements is NOT true?
Two obtuse angles each exceed 90°, so their sum exceeds 180° — they can never be supplementary. The other three statements are always true.
Question 11 of 30
What is the supplement of a 47° angle?
Supplementary angles add to 180°: 180 − 47 = 133°. The 43° is the complement (90 − 47); 313° subtracts from 360°.
Question 12 of 30
Two lines cross, and one of the angles measures 64°. What is the angle vertical to it?
Vertical angles, opposite through the crossing point, are equal: 64°. The 116° is the angle beside it, which forms a linear pair with it.
Question 13 of 30
Two angles are complementary, and one is 14° less than the other. What is the smaller angle?
Name the smaller x; the larger is x + 14. Then 2x + 14 = 90, so x = 38°. The 52° is the larger angle; 83° uses 180° instead of 90°.
Question 14 of 30
Which of these angles has a supplement but NO complement?
Only acute angles have complements: 90 − 104 is negative, so 104° has none, while its supplement is 76°. The 89° is still acute — its complement is 1°.
Question 15 of 30
Four angles meet around a point. Three of them measure 100°, 85° and 120°. What is the fourth?
Angles round a point total 360°: 360 − (100 + 85 + 120) = 360 − 305 = 55°. The 305° is the sum of the three given angles, not what is left.
Question 16 of 30
Two complementary angles measure (2x + 6)° and (3x − 1)°. What is the value of x?
Complementary: 5x + 5 = 90, so x = 17. The 35 uses 180, and 7 sets the two expressions equal (the vertical-angle rule). The 40 is one of the angles, 2(17) + 6.
Question 17 of 30
An angle is 4 times its supplement. What is the angle?
Call it A: A = 4(180 − A), so 5A = 720 and A = 144°. The 36° is its supplement, and 72° comes from using the complement, 90 − A.
Question 18 of 30
Two angles are complementary. One is 6° more than half the other. What is the larger angle?
Call the larger x; the other is x/2 + 6. Then x + x/2 + 6 = 90; multiply by 2: 3x + 12 = 180, so x = 56. The smaller is 34°. The 116° comes from using 180 as the total.
Question 19 of 30
Two lines cross. A 50° angle is vertical to an angle of (3x − y)° and forms a linear pair with an angle of (x + y)°. What is the value of x?
Vertical: 3x − y = 50. Linear pair: x + y = 180 − 50 = 130. Adding cancels y: 4x = 180, x = 45 (and y = 85). The 25 sets both expressions equal to 50.
Question 20 of 30
Three angles on a straight line measure x°, (x + 20)° and (2x − 4)°. What is the largest of them?
On a straight line they total 180: 4x + 16 = 180, so x = 41. The angles are 41°, 61° and 2(41) − 4 = 78°. Using 360 gives x = 86 and the 168°.
Question 21 of 30
The sum of two angles is 90° and their difference is 26°. What is the larger angle?
Name the smaller x: x + (x + 26) = 90, so x = 32 and the larger is 58°. Check: 58 + 32 = 90 and 58 − 32 = 26 ✓. The 32° is the smaller angle.
Question 22 of 30
The complement of an angle is 20° more than the angle. What is the supplement of the angle?
90 − A = A + 20 gives 2A = 70, so A = 35° and its supplement is 180 − 35 = 145°. The 35° is the angle, and 55° is its complement.
Question 23 of 30
The supplement of an angle is 10° less than 3 times its complement. What is the angle?
180 − A = 3(90 − A) − 10 = 260 − 3A, so 2A = 80 and A = 40°. Check: 140 = 3(50) − 10 ✓. The 50° and 140° are its complement and supplement.
Question 24 of 30
∠P = (5x − 12)° and ∠Q = (3x + 8)° are supplementary. What is the complement of ∠Q?
Supplementary: 8x − 4 = 180, so x = 23. ∠Q = 3(23) + 8 = 77°, and its complement is 90 − 77 = 13°. The 77° is ∠Q itself and 103° is ∠P.
Question 25 of 30
Three angles side by side on a straight line are in the ratio 2 : 3 : 4. What is the largest angle?
2k + 3k + 4k = 180, so 9k = 180 and k = 20. The largest is 4k = 80°. Using 360 gives 160°; the 20 is k, a multiplier, not an angle.
Question 26 of 30
Two lines cross. Three of the four angles they form add up to 250°. What does each of the two smaller angles measure?
All four total 360°, so the missing angle is 110°. Its vertical partner is also 110°, and each angle beside them is 180 − 110 = 70°. Check: 70 + 70 + 110 = 250 ✓.
Question 27 of 30
∠A and ∠B are supplementary, and ∠A is 3 times the complement of ∠B. What is ∠A?
A = 180 − B and A = 3(90 − B). So 180 − B = 270 − 3B, 2B = 90, B = 45° and A = 135°. Check: 3(90 − 45) = 135 ✓. The 45° is ∠B.
Question 28 of 30
Two vertical angles measure (5x − 20)° and (3x + 16)°. What is the measure of an angle that forms a linear pair with either of them?
Vertical: 5x − 20 = 3x + 16, so x = 18 and each angle is 70°. Its linear-pair partner is 180 − 70 = 110°. The 70° stops one step early; 140° adds the two vertical angles.
Question 29 of 30
Which statement is ALWAYS true?
If A < 90, then 180 − A > 90: obtuse. Adjacent only describes position (no rule by itself); complementary angles need not touch; vertical angles are equal, and complementary only when each is 45°.
Question 30 of 30
An acute angle measures x°. By how much does its supplement exceed its complement?
Supplement − complement = (180 − x) − (90 − x) = 90°, whatever x is. The (180 − 2x)° subtracts x twice instead of letting the two −x cancel.
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