Question

explain why the triangles are similar. then find the distance represented by x.90 ft180 ft120 ftx ftwhy 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 proportional, so the triangles are similar by the aa ~ postulate.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 a pair of congruent right angles, so the triangles are similar by the aa ~ postulate.d. 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 sas ~ theorem.e. the corresponding sides of two triangles are proportional, so the triangles are similar by the sss ~ theorem.$x=\square$ feet (type an integer or a simplified fraction.)

Explanation:

Step1: Verify triangle similarity

Both triangles have a right angle (congruent), and they share a pair of congruent vertical angles. By AA (Angle-Angle) Postulate, the triangles are similar.

Step2: Set up proportion

Corresponding sides of similar triangles are proportional:
$\frac{x}{180} = \frac{120}{90}$

Step3: Solve for x

Simplify the ratio and isolate x:
$x = 180 \times \frac{120}{90}$
$x = 180 \times \frac{4}{3}$
$x = 240$

Answer:

C. There is a pair of congruent vertical angles and a pair of congruent right angles, so the triangles are similar by the AA ~ Postulate.
$x=240$ feet