Problem 26 · 2011 Math Kangaroo
Stretch
Geometry & Measurement
casework
In a convex quadrilateral ABCD with AB = AC, the following holds true: ∠BAD = 80°, ∠ABC = 75°, ∠ADC = 65°. How big is ∠BDC? (Note: in a convex quadrilateral all internal angles are less than 180°.)
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Answer: B — 15°
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Hint 1 of 2
Use AB = AC to get the base angles of triangle ABC, then the quadrilateral angle sum.
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Hint 2 of 2
Show AB = AD, find ∠ADB, and subtract it from ∠ADC.
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Approach: chase angles using AB = AC and the quadrilateral angle sum
- AB = AC gives triangle ABC base angles 75°, so ∠BAC = 30° and ∠CAD = 50°.
- The quadrilateral angles sum to 360°, forcing ∠BCD = 140°, hence ∠ACD = 65°, and triangle ACD gives AC = AD.
- Then AB = AD too, so triangle ABD has base angles 50°, and ∠BDC = ∠ADC − ∠ADB = 65° − 50° = 15°.
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