Question

w - z 11th grade act math prep march 1
triangle sum = 180

1. in the figure below, $overline{ab}$ is congruent to $overline{bc}$, and $overline{ae}$ intersects $overline{bf}$ at c. what is the measure of $angle b$?

a. 14°
b. 38°
c. 76°
d. 104°
e. 142°

Explanation:

Step1: Find vertical - angle

The angle $\angle ACB$ and the given $38^{\circ}$ angle are vertical angles. So, $\angle ACB = 38^{\circ}$.

Step2: Use isosceles - triangle property

Since $\overline{AB}\cong\overline{BC}$, $\triangle ABC$ is isosceles. Let $\angle BAC=\angle BCA = 38^{\circ}$.

Step3: Apply triangle - sum theorem

In $\triangle ABC$, by the triangle - sum theorem, the sum of interior angles is $180^{\circ}$. Let $\angle B=x$. Then $x + 38^{\circ}+38^{\circ}=180^{\circ}$.
Solve for $x$: $x=180^{\circ}-(38^{\circ}+38^{\circ})=104^{\circ}$.

Answer:

D. $104^{\circ}$