# 2007 AMC 12A Problems/Problem 6

The following problem is from both the 2007 AMC 12A #6 and 2007 AMC 10A #8, so both problems redirect to this page.

## Problem

Triangles $ABC$ and $ADC$ are isosceles with $AB=BC$ and $AD=DC$. Point $D$ is inside triangle $ABC$, angle $ABC$ measures 40 degrees, and angle $ADC$ measures 140 degrees. What is the degree measure of angle $BAD$? $\mathrm{(A)}\ 20\qquad \mathrm{(B)}\ 30\qquad \mathrm{(C)}\ 40\qquad \mathrm{(D)}\ 50\qquad \mathrm{(E)}\ 60$

## Solution 1

We angle chase and find out that:

• $\angle DAC=\frac{180-140}{2} = 20$
• $\angle BAC=\frac{180-40}{2} = 70$
• $\angle BAD=\angle BAC- \angle DAC=50\ \mathrm{(D)}$

~minor edits by mobius247

## Solution 2

Since triangle $ABC$ is isosceles we know that angle $\angle BAC = \angle BCA$.

Also since triangle $ADC$ is isosceles we know that $\angle DAC = \angle DCA$.

This implies that $\angle BAD = \angle BCD$.

Then the sum of the interior angles of quadrilateral $ABCD$ is $40 + 220 + 2\angle BAD = 360$.

Solving the equation, we get $\angle BAD = 50$.

Therefore the answer is $\mathrm{(D)}$.

## Solution 3

Using the previous drawing we know that ACD and DAC are both equal to 20 (40/2 or (180-40)/2). We also know that BAC and BCA are both 70 or (180-40)/2. Thus BDA is 70-20 or 50. Easy. -RealityWrites

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 