Problem 29 · 2018 Math Kangaroo
Stretch
Geometry & Measurement
casework
In the isosceles triangle ABC (with base AC) the points K and L are added on the sides AB and BC respectively so that AK = KL = LB and KB = AC. How big is the angle ∠ABC?
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Answer: C — 36°
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Hint 1 of 2
Mark the equal segments and chase the base angles of the isosceles triangles that appear.
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Hint 2 of 2
Set the angle at A found two different ways equal to each other and solve for angle ABC.
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Approach: angle-chase the equal segments
- Let ∠ABC = β. From AK = KL = LB the angle at A inside that chain is β/2, so ∠LAC = (90° − β/2) − β/2 = 90° − β.
- Since KB = AC and B, L, C are in line, LC = AC, making triangle ALC isosceles with ∠LAC = 45° + β/4.
- Setting 90° − β = 45° + β/4 gives β = 36°.
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