Question 296·Hard·Math

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In triangles $PQR$ and $XYZ$ , $\angle P$ and $\angle X$ each measure $50^\circ$ , and $PQ = 8$ while $XY = 12$ . Which of the following additional pieces of information is sufficient to prove that $\triangle PQR$ is similar to $\triangle XYZ$ ?

A similarity question asking which fact is sufficient is asking what proves two triangles have the same shape. Match the vertices first, then look for a complete similarity rule. Two pairs of matching equal angles prove similarity by AA, even if the triangles have different side lengths. Extra side lengths alone work only if the matching ratios meet a similarity rule.

Hints

Hint 1
Vertices in the same positions in the triangle names are corresponding, or matching. Pair $P$ with $X$ , $Q$ with $Y$ , and $R$ with $Z$ before testing any added fact.
Hint 2
The AA similarity rule needs two pairs of matching equal angles. An angle stated for only one triangle doesn't give you a new pair. If a choice gives sides instead, their matching ratios must agree.

Step-by-step

Use a second matching angle
Step 1
Match the vertices
No Desmos needed. Equal angles can prove similarity directly. Vertices in the same positions in the triangle names are corresponding, or matching: $P\leftrightarrow X$ , $Q\leftrightarrow Y$ , and $R\leftrightarrow Z$ . So the given $50^\circ$ angles form one matching equal pair.
Step 2
Identify the angle rule
The AA similarity rule says two pairs of matching equal angles prove triangles are similar. You already have the pair at $P$ and $X$ , so you need one more equal pair. An angle given in only one triangle doesn't tell you its partner's measure.
Step 3
Use the second angle pair
The added fact $\angle Q=\angle Y=60^\circ$ supplies that second matching pair. By AA, $\triangle PQR\sim\triangle XYZ$ : they have the same shape, even though $PQ=8$ and $XY=12$ differ. Choice B.

Question 296·Hard·Math

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