Look at which choices add side lengths and which add angle measures. Which similarity rules use side lengths, and which use only angles?

For the choices that add side lengths, compare $\dfrac{PQ}{XY}$ with $\dfrac{QR}{YZ}$ or $\dfrac{PR}{XZ}$. Are those ratios equal? If not, they cannot give you SAS or SSS similarity.

Using the given $\angle P = \angle X = 50^\circ$ and each angle-based choice, ask: In how many triangles do I now know two corresponding angles with the same measure? That will tell you where AA similarity applies.

In Desmos, type 8/12, 13/18, and 9/14 on separate lines. Compare the decimal values: check whether 8/12 equals 13/18 (for option A) or 9/14 (for option C). If the ratios are not equal, those side-length conditions cannot give SAS or SSS similarity.

Type 180-50-70 to see what $\angle Q$ would be in $\triangle PQR$ if $\angle R = 70^\circ$ (as in option D). Notice that this still tells you only angles in $\triangle PQR$, not in both triangles, so it does not by itself create two equal corresponding angles.

For each angle-based choice, think in Desmos terms: how many corresponding angles between the two triangles become fixed to have the same measure? Use that to decide which condition creates AA similarity, without relying on side ratios.

To prove two triangles are similar, you can use:

• AA: Two corresponding angles are equal.
• SAS: Two pairs of corresponding sides are in the same ratio and the included angle is equal.
• SSS: All three pairs of corresponding sides are in the same ratio.

We are told $\angle P = \angle X = 50^\circ$ and $PQ = 8$, $XY = 12$ to start.

If a side-length option were enough, it would have to work with $PQ = 8$ and $XY = 12$ to give SAS similarity: two proportional sides with the included angle ($50^\circ$) equal.

• Option A gives $QR = 13$ and $YZ = 18$.
  • Compare $\dfrac{PQ}{XY} = \dfrac{8}{12}$ and $\dfrac{QR}{YZ} = \dfrac{13}{18}$.
  • $\dfrac{8}{12} = \dfrac{2}{3}$, but $\dfrac{13}{18}$ is not $\dfrac{2}{3}$, so the sides are not proportional.
• Option C gives $PR = 9$ and $XZ = 14$.
  • Compare $\dfrac{PQ}{XY} = \dfrac{8}{12}$ and $\dfrac{PR}{XZ} = \dfrac{9}{14}$.
  • $\dfrac{8}{12} = \dfrac{2}{3}$, but $\dfrac{9}{14}$ is not $\dfrac{2}{3}$, so these sides are not proportional either.

Because the side pairs are not in the same ratio, options A and C cannot prove the triangles are similar.

Now focus on the options that add angle measures:

• Option B gives $\angle Q$ and $\angle Y$.
• Option D gives $\angle R = 70^\circ$ (only in $\triangle PQR$).

For AA similarity, you need to know that two angles in $\triangle PQR$ match two angles in $\triangle XYZ$.

Option D only gives an angle in one triangle, so you would still know just one angle ($50^\circ$) in $\triangle XYZ$. That is not enough for AA similarity.

Option B directly tells you a pair of angles in $\triangle PQR$ and $\triangle XYZ$ have the same measure, which is exactly the kind of information AA needs.

We already know $\angle P = \angle X = 50^\circ$. If we also know from the additional information that $\angle Q = 60^\circ$ and $\angle Y = 60^\circ$, then:

• $\angle P$ in $\triangle PQR$ equals $\angle X$ in $\triangle XYZ$.
• $\angle Q$ in $\triangle PQR$ equals $\angle Y$ in $\triangle XYZ$.

That gives two pairs of equal corresponding angles, so $\triangle PQR$ is similar to $\triangle XYZ$ by AA similarity.

Therefore, the additional information "$\angle Q = 60^\circ$ and $\angle Y = 60^\circ$" is sufficient to prove similarity.