Problem 20 · 2013 Math Kangaroo
Medium
Geometry & Measurement
substitution
In triangle ABC the points M and N lie on side AB so that \(AN = AC\) and \(BM = BC\). Determine \(\angle ACB\) if \(\angle MCN = 43\degree\).
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Answer: E — 94°
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Hint 1 of 2
AN=AC and BM=BC make two isosceles triangles; chase the base angles.
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Hint 2 of 2
Express angle MCN using the triangle's angles A and B.
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Approach: isosceles angle chase
- In triangle ANC, angle ACN = 90° − A/2; in triangle BMC, angle BCM = 90° − B/2.
- Angle MCN = (90−A/2)+(90−B/2) − C = 90° − C/2.
- So 90 − C/2 = 43 gives C = 94°, choice E.
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