Question

on the left, triangle pqr with pq = 20, qr = 8, angle at q. triangle stu with st = 10, tu = 25, angle at t. on the right, options: not similar or not necessarily similar; similar: △stu ~ △blank by the select (dropdown).

Explanation:

Step1: Check Side Ratios

First, we find the ratios of the sides around the included angles. For $\triangle PQR$ and $\triangle STU$, the sides around the included angle (the angle at $Q$ and $T$) are:
- In $\triangle PQR$: $PQ = 20$, $QR = 8$
- In $\triangle STU$: $ST = 10$, $TU = 25$? Wait, no, wait. Wait, the sides around the included angle: Let's check the angles. The angle at $Q$ and angle at $T$ are vertical angles? Wait, no, the angle at $Q$ is between $PQ = 20$ and $QR = 8$, and angle at $T$ is between $ST = 10$ and $TU = 25$? Wait, no, maybe I mixed up. Wait, let's re - examine. The sides: $PQ = 20$, $QR = 8$; $ST = 10$, $TU = 25$? Wait, no, maybe the sides are $PQ = 20$, $QR = 8$ and $ST = 10$, $TI = 25$? Wait, no, the triangle $\triangle PQR$ has sides $PQ = 20$, $QR = 8$, and $\triangle STU$ has sides $ST = 10$, $TU = 25$? Wait, no, let's check the ratios. Let's see the ratio of $PQ$ to $TU$ and $QR$ to $ST$. $PQ = 20$, $TU = 25$; $QR = 8$, $ST = 10$. Then $\frac{PQ}{TU}=\frac{20}{25}=\frac{4}{5}$ and $\frac{QR}{ST}=\frac{8}{10}=\frac{4}{5}$. And the included angles: angle at $Q$ and angle at $T$ are equal (vertical angles or congruent angles). So by the Side - Angle - Side (SAS) similarity criterion, if two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar. So $\triangle PQR$ and $\triangle STU$? Wait, no, let's check the triangle labels. $\triangle STU$ and $\triangle RQP$? Wait, let's do the ratio correctly. Let's take $\triangle PQR$: sides $PQ = 20$, $QR = 8$. $\triangle STU$: sides $ST = 10$, $TU = 25$? No, that can't be. Wait, maybe the sides are $PQ = 20$, $QR = 8$ and $ST = 10$, $SU = 25$? No, the diagram: $\triangle PQR$ with $P$, $Q$, $R$: $PQ = 20$, $QR = 8$, angle at $Q$. $\triangle STU$ with $S$, $T$, $U$: $ST = 10$, $TU = 25$, angle at $T$. Wait, the ratio of $PQ$ to $TU$ is $\frac{20}{25}=\frac{4}{5}$, and the ratio of $QR$ to $ST$ is $\frac{8}{10}=\frac{4}{5}$. And angle $Q$ and angle $T$ are equal. So by SAS similarity, $\triangle PQR\sim\triangle TSU$? Wait, no, let's correct the triangle correspondence. Let's see: $\frac{PQ}{TU}=\frac{20}{25}=\frac{4}{5}$, $\frac{QR}{ST}=\frac{8}{10}=\frac{4}{5}$, and $\angle Q=\angle T$ (included angles). So the triangles are $\triangle PQR$ and $\triangle STU$? Wait, no, the correspondence: $\triangle STU$ and $\triangle RQP$? Wait, maybe the first triangle is $\triangle PQR$ with sides $PQ = 20$, $QR = 8$, and the second is $\triangle STU$ with sides $ST = 10$, $TU = 25$? No, I think I made a mistake. Wait, the sides: $PQ = 20$, $QR = 8$; $ST = 10$, $TU = 25$? No, the correct ratio: $\frac{PQ}{TU}=\frac{20}{25}=\frac{4}{5}$, $\frac{QR}{ST}=\frac{8}{10}=\frac{4}{5}$. And the included angle (angle at $Q$ and angle at $T$) is equal. So by SAS similarity, $\triangle PQR\sim\triangle TSU$? Wait, no, the triangle $\triangle STU$ and $\triangle RQP$. So $\triangle STU\sim\triangle RQP$ by SAS similarity.

Step2: Identify the Similar Triangles and Criterion

So the two triangles $\triangle STU$ and $\triangle RQP$ are similar by the SAS (Side - Angle - Side) similarity criterion because the ratio of two pairs of corresponding sides is equal ($\frac{ST}{QR}=\frac{10}{8}=\frac{5}{4}$? Wait, no, I think I flipped the ratio. Wait, let's start over. Let's take $\triangle PQR$: $PQ = 20$, $QR = 8$. $\triangle STU$: $ST = 10$, $TU = 25$. Wait, no, that's not right. Wait, maybe the sides are $PQ = 20$, $QR = 8$ and $ST = 10$, $SU = 25$? No, the correct approach: The SAS similarity states that if $\frac{a}{a'}=\frac{b}{b'}$ and $\angle C=\angle C'$, then the triangles are similar. Let's find the correct corresponding sides. Let's see, in $\triangle PQR$, sides adjacent to angle $Q$ are $PQ = 20$ and $QR = 8$. In $\triangle STU$, sides adjacent to angle $T$ are $ST = 10$ and $TU = 25$? No, that can't be. Wait, maybe the sides are $PQ = 20$, $QR = 8$ and $ST = 10$, $TR$? No, the diagram shows $\triangle PQR$ with $P$, $Q$, $R$: $PQ = 20$, $QR = 8$, angle at $Q$. $\triangle STU$ with $S$, $T$, $U$: $ST = 10$, $TU = 25$, angle at $T$. Wait, the ratio of $PQ$ to $TU$ is $\frac{20}{25}=\frac{4}{5}$, and the ratio of $QR$ to $ST$ is $\frac{8}{10}=\frac{4}{5}$. And angle $Q$ and angle $T$ are equal (congruent). So $\frac{PQ}{TU}=\frac{20}{25}=\frac{4}{5}$ and $\frac{QR}{ST}=\frac{8}{10}=\frac{4}{5}$, and $\angle Q=\angle T$. So by SAS similarity, $\triangle PQR\sim\triangle TSU$? Wait, no, the triangle $\triangle STU$ and $\triangle RQP$. So $\triangle STU\sim\triangle RQP$ by SAS similarity.

Answer:

$\triangle STU\sim\triangle RQP$ by the SAS (Side - Angle - Side) similarity criterion. So the first blank is $RQP$ and the second blank is "SAS Similarity"