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

For two triangles to be similar, we can use the AA (angle - angle), SAS (side - angle - side), SSS (side - side - side) similarity criteria.

Step2: Analyze given information

In \(\triangle ABC\) and \(\triangle PQR\), we know that \(\angle A=\angle P = 58^{\circ}\), \(AB = 35\), \(PQ = 35\), \(AC = 30\), \(PR=120\), \(QR = 140\).
We check the ratio of corresponding sides. \(\frac{AB}{PQ}=\frac{35}{35} = 1\), \(\frac{AC}{PR}=\frac{30}{120}=\frac{1}{4}\), \(\frac{AB}{PQ}
eq\frac{AC}{PR}\).
We also note that just knowing one pair of equal angles (\(\angle A=\angle P\)) and the given side - lengths does not satisfy any of the similarity criteria (AA: we need two pairs of equal angles; SAS: we need the ratio of two pairs of sides and the included angle equal; SSS: we need the ratio of all three pairs of sides equal).

Answer:

No, the information is not sufficient to prove that \(\triangle ABC\) is similar to \(\triangle PQR\).