In triangle ABC, AB = AC. Point D lies on BC so that AD = DC. If the measure of angle BAD is 75^, what is the measure of angle BAC?

In triangle ABC, AB = AC. Point D lies on BC so that AD = DC... | SAT Lines, angles, and triangles Question, Answer Explained

Question bank / Lines, angles, and triangles

Math · Geometry and TrigonometryHard

In triangle $ABC$ , $AB = AC$ . Point $D$ lies on $\overline{BC}$ so that $AD = DC$ . If the measure of angle $BAD$ is $75^\circ$ , what is the measure of angle $BAC$ ?

$35^\circ$

$70^\circ$

$110^\circ$

$150^\circ$

Show answer and explanation

Answer: C

Let the measure of angle $C$ be $x^\circ$ . Because $AB = AC$ , angle $B$ also measures $x^\circ$ . Because $AD = DC$ , triangle $ADC$ is isosceles, so angle $DAC$ also measures $x^\circ$ . Then angle $BAC$ measures $(75 + x)^\circ$ , and the angles of triangle $ABC$ give $(75 + x) + x + x = 180$ . So $3x = 105$ , $x = 35$ , and angle $BAC$ measures $75^\circ + 35^\circ = 110^\circ$ .

Choice D assumes that $\overline{AD}$ bisects angle $BAC$ , so that angle $DAC$ is also $75^\circ$ , but nothing in the given information says so.

Why the other choices are wrong

A. 35 degrees is the measure of angle B or angle C, not angle BAC.
B. 70 degrees is the measure of angle ADB, an exterior angle of triangle ADC.
D. This assumes AD bisects angle BAC, but nothing says so. Angle DAC is 35 degrees.

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