Question

question 23 of 25 according to the diagram below, which similarity statements are true? check all that apply. a. △pqr~△psq b. △pqr~△prs c. △pqr~△qsr d. △pqs~△qrs

Explanation:

Step1: Recall similarity criteria

In right - angled triangles, if an altitude is drawn from the right - angle vertex to the hypotenuse, then the three right - angled triangles formed are similar to each other. In right - triangle $PQS$ with altitude $QR$ from right - angle $Q$ to hypotenuse $PS$.

Step2: Analyze option A

In $\triangle PQR$ and $\triangle PSQ$, $\angle P$ is common and $\angle PQR=\angle PSQ = 90^{\circ}-\angle QPR$. By the AA (angle - angle) similarity criterion, $\triangle PQR\sim\triangle PSQ$.

Step3: Analyze option B

$\triangle PQR$ and $\triangle PRS$ are not similar. $\triangle PQR$ is a right - triangle with right - angle at $Q$ and $\triangle PRS$ is not a right - triangle in the same sense as $\triangle PQR$.

Step4: Analyze option C

In $\triangle PQR$ and $\triangle QSR$, $\angle QRP=\angle SRQ = 90^{\circ}$ and $\angle PQR = 90^{\circ}-\angle RQS$, $\angle S=90^{\circ}-\angle RQS$, so $\angle PQR=\angle S$. By the AA similarity criterion, $\triangle PQR\sim\triangle QSR$.

Step5: Analyze option D

In $\triangle PQS$ and $\triangle QRS$, $\angle PQS=\angle QRS = 90^{\circ}$ and $\angle S$ is common. By the AA similarity criterion, $\triangle PQS\sim\triangle QRS$.

Answer:

A. $\triangle PQR\sim\triangle PSQ$
C. $\triangle PQR\sim\triangle QSR$
D. $\triangle PQS\sim\triangle QRS$