Problem 9 · 2014 AMC 8
Easy
Geometry & Measurement
isosceles-trianglelinear-pair
In ▵ABC, D is a point on side AC such that BD = DC and ∠BCD measures 70°. What is the degree measure of ∠ADB?
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Answer: D — 140 degrees.
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Hint 1 of 2
BD = DC makes ▵BDC isosceles, so the 70° at C is mirrored at B — you instantly get a second 70° for free.
Still stuck? Show hint 2 →
Hint 2 of 2
∠ADB sits on the straight line AC next to ▵BDC, so it's the exterior angle — equal to the sum of the two far (remote) interior angles.
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Approach: exterior angle of an isosceles triangle
- BD = DC ⇒ ▵BDC is isosceles, so its base angles match: ∠DBC = ∠DCB = 70°. (Equal sides face equal angles.)
- ∠ADB is the exterior angle of ▵BDC at D. The exterior angle equals the sum of the two remote interior angles: 70° + 70° = 140°.
- Why this transfers: the exterior-angle rule (exterior = sum of the two non-adjacent interiors) lets you skip finding the apex angle entirely — it's a shortcut worth spotting whenever an angle hangs off a straight side.
Another way — linear pair after computing the apex angle:
- Base angles 70° each give apex ∠BDC = 180° − 140° = 40°. Then ∠ADB = 180° − 40° = 140°.
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