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 angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the sas ~ theorem.
b. there is a pair of congruent angles and the sides that include the two vertical angles are proportional, so the triangles are similar by the aa ~ postulate.
c. the corresponding sides of two triangles are proportional, so the triangles are similar by the sss ~ theorem
d. there is a pair of congruent angles and a pair of congruent right angles, so the triangles are similar by the aa ~ postulate.
e. there is a pair of congruent angles and the sides that include the two vertical angles are also congruent, so the triangles are similar by the sas ~ theorem.
x = \square feet (simplify your answer. type an integer or a decimal.)

Explanation:

Step1: Verify triangle similarity

Both triangles are right triangles (one right angle each), and the non-right angles at the intersection are vertical angles, so they are congruent. By AA (Angle-Angle) Postulate, the triangles are similar.

Step2: Convert units to feet

The height of the smaller triangle is 4 ft 6 in. Since 6 in = $\frac{6}{12}$ ft = 0.5 ft, the height is $4.5$ ft.

Step3: Set up proportion for similar triangles

For similar triangles, corresponding sides are proportional:
$$\frac{x}{4.5} = \frac{42}{12}$$

Step4: Solve for x

Multiply both sides by 4.5:
$$x = 4.5 \times \frac{42}{12}$$
$$x = 4.5 \times 3.5$$
$$x = 15.75$$

Answer:

D. There is a pair of congruent angles and a pair of congruent right angles, so the triangles are similar by the AA - Postulate.
$x = 15.75$ feet