To prove that the triangles are similar by the SAS similarity theorem, it needs to be proven that two sides and included angles in both are congruent?

Solution:

SAS Similarity theorem states that, “If two sides in one triangle are proportional to two sides in another triangle and the included angle in both are congruent, then the two triangles are similar”.

Proof:

Given: PQ/XY = QR/YZ and ∠Q ≅ ∠Y

To prove that, △PQR is similar to △XYZ.

Proof: Assume PQ > XY

Draw MN parallel to BC, we find that MQN similar to XYZ

QM/QP = QN/QR --- (1)[using the basic proportionality theorem]

Now △MQN and △XYZ are congruent thus, XY/QP = YZ/QR --- (2)

Since QM = XY from (1) and (2),

XY/QP = QM/QP = QN/QR = YZ/QR

Thus, QN = YZ by SAS congruence criterion.

△MQN ≅ △XYZ

But △MQN congruent to △XYZ,

Thus, △PQR is similar to △XYZ.

To prove that the triangles are similar by the SAS similarity theorem, it needs to be proven that two sides and included angles in both are congruent?

Summary:

To prove that the triangles are similar by the SAS similarity theorem, it needs to be proven that two sides and included angles in both are congruent.