Problem 26 · 2015 Math Kangaroo
Stretch
Geometry & Measurement
substitution
In the trapezium PQRS the sides PQ and SR are parallel. Also \(\angle RSP = 120^\circ\) and \(RS = SP = \tfrac{1}{3}PQ\). What is the size of angle \(\angle PQR\)?
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Answer: D — 30°
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Hint 1 of 2
Drop the equal sides RS = SP = (1/3)PQ into the trapezium and use that PQ is parallel to SR.
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Hint 2 of 2
The isosceles pieces and the 120 degree angle let you chase angles down to angle PQR.
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Approach: angle-chase using the equal sides and parallels
- RS = SP makes triangle RSP isosceles, and angle RSP = 120 gives base angles of 30 degrees.
- Since RS = SP = PQ/3, the equal lengths split the figure into congruent isosceles triangles with 30 degree base angles.
- Chasing these angles across to the corner Q gives angle PQR = 30 degrees.
- So angle PQR = 30 degrees (D).
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