Problem 13 · 2009 Math Kangaroo
Medium
Geometry & Measurement
sum-constraintcasework
Nick measured all 6 angles in two triangles. One of the triangles was acute-angled and the other obtuse-angled. He noted four of the angles to be 120°, 80°, 55° and 10°. What is the size of the smallest angle in the acute-angled triangle?
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Answer: A — 45°
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Hint 1 of 2
The 120 degree angle must belong to the obtuse triangle; the acute triangle's angles are all below 90.
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Hint 2 of 2
Split the four given angles so one triangle is obtuse and the other has three angles under 90 summing to 180.
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Approach: sort the angles into the two triangles
- The obtuse triangle holds 120, leaving 60 for its other two angles, e.g. 10 and 50.
- The acute triangle then uses 80 and 55, needing a third angle of 180 - 80 - 55 = 45 (all under 90 - valid).
- The smallest angle in the acute triangle is 45 degrees.
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