AMC 10 · 2002 · #4

Grade 7 geometry-2d
Archetype Multiplicative Ratio with Integrality Constraints └ Sub-archetype Direct Ratio Scale-Up
complementary-anglessupplementary-angleslinear-equations-one-var convert-to-algebra ↑ Prerequisites: complementary-anglessupplementary-angles
📏 Medium solution 💡 2 insights
Problem
An angle has some degree measure. Its complement — what is left of 90° after taking the angle away — turns out to be 25% of its supplement, which is what is left of 180° after taking the same angle away. Find the degree measure of the angle.

Pick an answer.

(A)
48
(B)
60
(C)
75
(D)
120
(E)
150
How to solve
Strategy Convert to Algebra

The whole problem is one English sentence about one unknown angle, and every phrase in it ("complement", "supplement", "is", "25% of") has an exact algebraic translation. Tool #13 (Convert to Algebra) turns that sentence into a single linear equation. Tool #4 (Introduce a Variable) does the naming that makes the translation possible: once the angle is called x, the complement and supplement stop being vocabulary and become the expressions 90 - x and 180 - x. Tool #6 (Guess and Check) is then used the honest way — not to search for the answer, but to substitute the solved value back into the original sentence. The trap to avoid is stopping one step early and reporting the supplement instead of the angle.

1STEP 1

Name the angle and its partners

Let x be the angle; then the complement is 90 - x and the supplement is 180 - x.

complement = 90 - x, supplement = 180 - x
2STEP 2

Turn the percent into an equation

25% is one quarter, so 90 - x = 1/4(180 - x); multiply by 4 to get 4(90 - x) = 180 - x.

90 - x = 1/4(180 - x) ⟺ 4(90 - x) = 180 - x
3STEP 3

Solve for the angle

Expand to 360 - 4x = 180 - x, collect to 180 = 3x, so x = 60.

360 - 4x = 180 - x → 180 = 3x → x = 60
4STEP 4

Check it against the sentence

Check: complement 30 is a quarter of supplement 120, so 60 is right, choice (B).

90 - 60 = 30, 180 - 60 = 120, 1/4 × 120 = 30
Answer
60
Two checks agree. Direct substitution: at 60° the complement is 30°, the supplement is 120°, and 30 is exactly a quarter of 120. Structural check: for any angle, the supplement is always exactly 90° larger than the complement, since (180 - x) - (90 - x) = 90; here 120 - 30 = 90, as required. The distractors are also readable: 120 (choice D) is this angle's supplement, and 150 (choice E) is 180 - 30, the supplement of the complement. Both of those are larger than 90°, so neither could be the answer in the first place — an angle above 90° has no complement to talk about.
💡Key takeaway

Name the unknown angle, write what "complement" and "supplement" mean in terms of that name, and the sentence turns into an equation you can just solve — then re-read the question to make sure you report the angle, not its supplement.

  • Name the angle and its partners
  • Turn the percent into an equation
  • Solve for the angle
  • Check it against the sentence

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