Question

Quadrilateral $ABCD$ has $AB=BC=CD,$ $\angle ABC=70^\circ,$ and $\angle BCD=170^\circ.$ What is the degree measure of $\angle BAD?$

$75$

$80$

$85$

$90$

$95$

Answer: C

Concepts:angle chasing isosceles triangle equilateral triangle

Difficulty rating: 1860

Solution:

Let $M$ be the point with $\triangle BMC$ equilateral, on the same side of $BC$ as $A.$ Then $\angle ABM=70^\circ-60^\circ=10^\circ$ and $\angle MCD=170^\circ-60^\circ=110^\circ.$

Since $AB=BM$ and $MC=CD,$ triangles $ABM$ and $MCD$ are isosceles, giving $\angle AMB=85^\circ$ and $\angle CMD=35^\circ.$

Then $\angle AMD=360^\circ-85^\circ-60^\circ$ $-35^\circ$ $=180^\circ,$ so $M$ lies on $\overline{AD}$ and $\angle BAD=\angle BAM=85^\circ.$

Thus, the correct answer is C.

2008 AMC 10B Problem 24

Answer: C

Difficulty rating: 1860

Solution:

Problem 24 in Other Years