Question

explain why the triangles are similar. then find the distance represented by x.
why are the triangles similar? choose the correct answer below.
a. there is a pair of congruent vertical angles and the sides that include the two vertical angles are also congruent, so the triangles are similar by the sas - theorem.
b. there is a pair of congruent vertical angles and the sides that include the two vertical angles are also congruent, so the triangles are similar by the sas - theorem.
c. there is a pair of congruent vertical angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the aa - postulate.
d. the corresponding sides of two triangles are proportional, so the triangles are similar by the sss - theorem.
e. there is a pair of congruent vertical angles and a pair of congruent right angles, so the triangles are similar by the aa - postulate.

Explanation:

Step1: Identify similar - triangle condition

There is a pair of congruent vertical angles and a pair of congruent right - angles. By the AA (Angle - Angle) postulate, the triangles are similar.

Step2: Set up proportion

Since the triangles are similar, the ratios of corresponding sides are equal. We have the proportion $\frac{180}{90}=\frac{120}{x}$.

Step3: Cross - multiply

Cross - multiplying gives us $180x = 90\times120$.

Step4: Solve for x

First, calculate $90\times120 = 10800$. Then, $x=\frac{10800}{180}=60$.

Answer:

E. There is a pair of congruent vertical angles and a pair of congruent right angles, so the triangles are similar by the AA - Postulate.
$x = 60$ ft