Problem 8 · 2009 Math Kangaroo
Easy
Geometry & Measurement
symmetry
In the diagram QSR is a straight line. ∠QPS = 12° and PQ = PS = RS. How big is ∠QPR?
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Answer: C — 54°
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Hint 1 of 2
Two isosceles triangles share the straight line QSR; find each base angle in turn.
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Hint 2 of 2
Use PQ=PS, then PS=RS, and add the two angles at P.
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Approach: chase angles through two isosceles triangles
- In triangle PQS, PQ=PS with apex angle QPS=12, so each base angle is (180-12)/2 = 84; in particular angle PSQ = 84.
- Since QSR is straight, angle PSR = 180 - 84 = 96. In triangle PSR, PS=RS, so angle SPR = (180-96)/2 = 42.
- Then angle QPR = 12 + 42 = 54 degrees.
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