Question

decide whether the triangles are similar. if they are, write a similarity statement and state the reason justifying the similarity. if necessary, you may learn what the markings on a figure indicate.
(image of three triangle pairs with multiple - choice options for similarity determination)

Explanation:

Step1: Analyze First Triangle Pair

• Check angles: $\angle Y$ and $\angle N$ are right angles ($90^\circ$), so $\angle Y = \angle N$. Also, $\angle XZY$ and $\angle MZN$ are vertical angles, so $\angle XZY = \angle MZN$. By AA (Angle - Angle) similarity criterion, $\triangle XYZ \sim \triangle MNZ$.

Step2: Analyze Second Triangle Pair

• Calculate side ratios: For $\triangle DEF$ and $\triangle ABC$, sides of $\triangle DEF$: $DE = 5$, $DF = 9$, $EF = 7$; sides of $\triangle ABC$: $AB = 14$, $BC = 10$, $AC = 18$. Check ratios: $\frac{DE}{BC}=\frac{5}{10}=\frac{1}{2}$, $\frac{EF}{AB}=\frac{7}{14}=\frac{1}{2}$, $\frac{DF}{AC}=\frac{9}{18}=\frac{1}{2}$. All ratios are equal, so by SSS (Side - Side - Side) similarity criterion, $\triangle ABC \sim \triangle EFD$.

Step3: Analyze Third Triangle Pair

• Check angles: $\angle Q$ and $\angle T$ are equal (marked as equal). Calculate side ratios: For $\triangle PQR$: $PQ = 16$, $QR = 12$; for $\triangle STU$: $ST = 15$, $TU = 20$. Ratios: $\frac{PQ}{TU}=\frac{16}{20}=\frac{4}{5}$, $\frac{QR}{ST}=\frac{12}{15}=\frac{4}{5}$. By SAS (Side - Angle - Side) similarity criterion (since included angle is equal and sides are proportional), $\triangle STU \sim \triangle RQP$ (or $\triangle PQR$ depending on correspondence, but here $\triangle STU \sim \triangle RQP$ as $\frac{ST}{QR}=\frac{15}{12}=\frac{5}{4}$ and $\frac{TU}{PQ}=\frac{20}{16}=\frac{5}{4}$, wait correction: Wait, $\triangle PQR$ has sides $PQ = 16$, $QR = 12$, $\triangle STU$ has $ST = 15$, $TU = 20$. Wait, $\angle Q$ and $\angle T$ are equal. So $\frac{PQ}{TU}=\frac{16}{20}=\frac{4}{5}$, $\frac{QR}{ST}=\frac{12}{15}=\frac{4}{5}$. So $\triangle PQR \sim \triangle TUS$? Wait no, let's re - check. $\triangle STU$: $ST = 15$, $TU = 20$, $\angle T$; $\triangle PQR$: $PQ = 16$, $QR = 12$, $\angle Q$. So $\frac{PQ}{TU}=\frac{16}{20}=\frac{4}{5}$, $\frac{QR}{ST}=\frac{12}{15}=\frac{4}{5}$, and $\angle Q=\angle T$. So by SAS, $\triangle PQR \sim \triangle TUS$? Wait the problem has $\triangle STU \sim \triangle$? Wait the third triangle: $\triangle STU$ with $ST = 15$, $TU = 20$, $\angle T$; $\triangle PQR$ with $PQ = 16$, $QR = 12$, $\angle Q$. So the similarity is $\triangle STU \sim \triangle RQP$? Wait no, let's do ratios properly. $\frac{ST}{QR}=\frac{15}{12}=\frac{5}{4}$, $\frac{TU}{PQ}=\frac{20}{16}=\frac{5}{4}$, and $\angle T=\angle Q$. So by SAS, $\triangle STU \sim \triangle RQP$ (since $ST$ corresponds to $QR$, $TU$ corresponds to $PQ$, and included angle $\angle T$ and $\angle Q$).

Answer:

1. First Pair: Similar: $\triangle XYZ \sim \triangle MNZ$ by AA Similarity.
2. Second Pair: Similar: $\triangle ABC \sim \triangle EFD$ by SSS Similarity.
3. Third Pair: Similar: $\triangle STU \sim \triangle RQP$ by SAS Similarity. (Note: The exact correspondence may need to be adjusted based on precise angle - side correspondence, but the key is the similarity criterion application.)