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 - criteria

Two triangles are similar if two - angles of one triangle are equal to two - angles of another triangle (AA similarity criterion).

Step2: Analyze option a

If \(AB = 35\) and \(PQ = 35\), and \(\angle A=\angle P = 58^{\circ}\), \(AC = 30\), \(PR = 120\), the ratio of sides \(\frac{AB}{PQ}=1\) and \(\frac{AC}{PR}=\frac{30}{120}=\frac{1}{4}\). Sides are not in proportion, so not similar.

Step3: Analyze option b

Given \(AB = 35\), \(QR = 140\), \(\angle A=\angle P = 58^{\circ}\), \(AC = 30\), \(PR = 120\). We don't have enough information about the relationship between the sides and angles to prove similarity.

Step4: Analyze option c

If \(\angle B = 34^{\circ}\) and \(\angle R=88^{\circ}\), in \(\triangle ABC\), \(\angle C=180^{\circ}-\angle A - \angle B=180 - 58-34 = 88^{\circ}\). In \(\triangle PQR\), \(\angle Q=180^{\circ}-\angle P-\angle R=180 - 58 - 88=34^{\circ}\). Since \(\angle A=\angle P = 58^{\circ}\), \(\angle B=\angle Q = 34^{\circ}\), and \(\angle C=\angle R = 88^{\circ}\), by AA similarity criterion, \(\triangle ABC\sim\triangle PQR\).

Step5: Analyze option d

If \(\angle B = 58^{\circ}\) and \(\angle Q = 34^{\circ}\), \(\angle A=\angle P = 58^{\circ}\), but we don't have enough information to prove similarity as the angle - angle correspondence is not correct for similarity.

Answer:

c. the measures of angle B and angle R are 34 and 88 respectively