Problem 5 · 1988 AJHSME
Medium
Geometry & Measurement
protractor-readingangle-subtraction
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Answer: C — 50°.
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Hint 1 of 2
A protractor doesn't measure an angle directly — it gives each ray a number, and the angle *between* two rays is the difference of their two numbers. So first find the number on every ray you care about.
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Hint 2 of 2
C has no label, but ∠CBD = 90° pins it down: C's reading must be 90° less than D's. Once you know C and A's readings, ∠ABC is just C − A.
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Approach: an angle = difference of the two protractor readings
- Read each ray off the scale: A is at 20° and D is at 160°. The angle between any two rays is the gap between their readings — that's why ∠CBD = (C's reading) − (D's reading) would be negative, so we use the 90° gap the other way: C sits 90° below D, at 160° − 90° = 70°.
- Now ∠ABC = (C's reading) − (A's reading) = 70° − 20° = 50°.
- Why this transfers: think of a protractor as a number line bent into a half-circle. Any angle is a subtraction of two positions, just like distance on a ruler is a subtraction of two marks.
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