Question

in \\( \triangle xyz \\) and \\( \triangle pqr \\), if \\( \angle x = \angle p, \angle y = \angle q \\), and side lengths satisfy \\( \frac { x y } { p q } = \frac { y z } { q r } = \frac { x z } { p r } \\), which of the following is true?
a. the triangles are congruent.
b. the triangles are similar by sss similarity.
c. the triangles are similar by aa similarity.
d. the triangles are not similar.

Explanation:

Step1: Recall the SSS similarity criterion

If in two triangles, the corresponding sides are in proportion, then the triangles are similar. Here, we have \(\frac{XY}{PQ}=\frac{YZ}{QR}=\frac{XZ}{PR}\), which satisfies the SSS (Side - Side - Side) similarity criterion.

Step2: Analyze other options

• Option a: Congruent triangles require \(XY = PQ\), \(YZ=QR\), \(XZ = PR\) (SSS congruence). But we are only given proportionality, not equality.
• Option c: AAA similarity requires equality of corresponding angles. We are given side - side ratios.
• Option d: Since SSS similarity is satisfied, this option is wrong.

Answer:

B. The triangles are similar by SSS similarity.