Question

according to the diagram below, which similarity statements are true? check all that apply: a. $\triangle pqr \sim \triangle qsr$ b. $\triangle pqr \sim \triangle prs$ c. $\triangle pqs \sim \triangle qrs$ d. $\triangle pqr \sim \triangle psq$

Explanation:

Step1: Identify right angles

$\angle PRQ = \angle QRS = \angle PQS = 90^\circ$

Step2: Check shared angles for A

In $\triangle PQR$ and $\triangle QSR$:
$\angle PRQ = \angle QRS = 90^\circ$, $\angle P = \angle SQR$ (complementary to $\angle PQR$). By AA similarity, $\triangle PQR \sim \triangle QSR$.

Step3: Check shared angles for D

In $\triangle PQR$ and $\triangle PSQ$:
$\angle PRQ = \angle PQS = 90^\circ$, $\angle P$ is common. By AA similarity, $\triangle PQR \sim \triangle PSQ$.

Step4: Eliminate incorrect options

Option B: $\triangle PRS$ is not a right triangle, no matching angles. Option C: $\triangle PQS$ is large right triangle, $\triangle QRS$ is smaller, no matching angle pairs besides right angle.

Answer:

A. $\triangle PQR \sim \triangle QSR$
D. $\triangle PQR \sim \triangle PSQ$