Question

in triangle abc, the measure of angle a is 58 and ac = 30. in triangle pqr, the measure of angle p is 58 and pr = 120. which additional piece of information is sufficient to prove that triangle abc is similar to triangle pqr? a. ab = 35 and pq = 35 b. ab = 35 and qr = 140 c. the measures of angle b and angle r are 34 and 88 respectively d. the measures of angle b and angle q are 58 and 34 respectively

Explanation:

Step1: Recall similarity criterion

For two triangles to be similar, we can use the Side - Angle - Side (SAS) similarity criterion which states that if the ratios of two pairs of corresponding sides are equal and the included angles are congruent, the triangles are similar.

Step2: Check side - side ratio

In option a, we have $\frac{AB}{PQ}=\frac{35}{35} = 1$ and $\frac{AC}{PR}=\frac{30}{120}=\frac{1}{4}$. The ratios of the corresponding sides are not equal.
In option b, we have $\frac{AB}{PQ}=\frac{35}{35} = 1$ and $\frac{AC}{PR}$ (not given for this option in terms of side - side comparison). But just with $AB = 35,PQ = 35,QR=140$ we cannot prove similarity.
In option c, we are given angle measures. But just knowing angle measures of two non - corresponding angles does not prove similarity on its own.
In option d, we know that $\angle B=\angle Q = 58^{\circ}$ and $\angle C=\angle R=34^{\circ}$. By the Angle - Angle (AA) similarity criterion (if two angles of one triangle are equal to two angles of another triangle, the two triangles are similar), this information is sufficient to prove that $\triangle ABC\sim\triangle PQR$.

Answer:

d. the measures of angle B and angle Q are 58 and 34 respectively