Triangles ABC and ADC are isosceles with AB=BC and AD=DC. Point D is inside \triangle ABC, \angle ABC=40°, and \angle ADC=140°. What is the degree measure of \angle BAD?

2007 AMC 12A Problem 6 — Solution

2007 AMC 12A Problem 6

Below is the professionally curated solution for Problem 6 of the 2007 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 12A solutions, or check the answer key.

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Concepts:isosceles triangle angle chasing

Difficulty rating: 1200

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC.$ Point $D$ is inside $\triangle ABC,$ $\angle ABC=40^\circ,$ and $\angle ADC=140^\circ.$ What is the degree measure of $\angle BAD?$

$20$

$30$

$40$

$50$

$60$

Solution:

Since $\triangle ABC$ is isosceles, $\angle BAC=\tfrac12(180^\circ-\angle ABC)=70^\circ.$

Since $\triangle ADC$ is isosceles, $\angle DAC=\tfrac12(180^\circ-\angle ADC)=20^\circ.$

Therefore $\angle BAD=\angle BAC-\angle DAC=70^\circ-20^\circ=50^\circ.$

Thus, the correct answer is D.

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