Problem 13 · 2010 Math Kangaroo
Medium
Geometry & Measurement
symmetry
In the quadrilateral ABCD, AD = BC, \(\angle DAC = 50^\circ\), \(\angle DCA = 65^\circ\), and \(\angle ACB = 70^\circ\). How big is \(\angle ABC\)?
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Answer: B — 55°
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Hint 1 of 2
Work inside triangle ACD first using the given angles.
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Hint 2 of 2
AD = BC and the shared side AC let you compare triangles ACD and ACB.
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Approach: angle-chase with the equal sides
- In triangle ACD: ∠DAC = 50°, ∠DCA = 65°, so ∠ADC = 65°; thus AC = AD.
- Since AD = BC, we get AC = BC, so triangle ACB is isosceles with ∠ACB = 70°.
- Then ∠ABC = ∠BAC = (180°−70°)/2 = 55°.
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