Problem 23 · 2023 Math Kangaroo
Stretch
Geometry & Measurement
substitution
The triangle ABC shown is isosceles with ∠ABC = 40°. The two indicated angles ∠EAB and ∠DCA are equal. How big is the angle ∠CFE?
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Answer: D — 70°
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Hint 1 of 2
Use that triangle ABC is isosceles to get its base angles from the 40° apex.
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Hint 2 of 2
Let the equal marked angles be x; chase angles through the triangle CFE using the exterior-angle idea.
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Approach: angle chase using the isosceles base angles and the equal marked angles
- With apex angle ABC = 40°, the base angles satisfy ∠BAC = ∠BCA = (180 − 40)/2 = 70°.
- Let the equal angles be ∠EAB = ∠DCA = x. In triangle AFC, ∠FAC = 70° − x and ∠FCA = x.
- So ∠AFC = 180° − (70° − x) − x = 110° — the x cancels.
- ∠CFE is the supplement of ∠AFC: 180° − 110° = 70°, which is option D.
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