Shown below are three triangles. The measures of two adjacent sides and included angle are given for each triangle : Which of these triangles are similar?
Show Hint
When checking for SAS similarity, always ensure that:
1. The given angle is strictly the "included" angle between the two specified sides.
2. You compare the ratios of corresponding sides (i.e., compare the longer side of one triangle with the longer side of the other, and the shorter with the shorter).
$\Delta RPQ$, $\Delta XZY$ and $\Delta MNL$ are similar to one another
Show Solution
The Correct Option isA
Solution and Explanation
Step 1: Compute all three side ratios first.
Instead of comparing two triangles at a time, let us find the ratio of the two given sides for all three triangles up front, then match them in one pass. Recall the SAS similarity rule: two triangles are similar if one pair of included angles is equal and the sides forming that angle are in the same ratio.
Step 2: Ratio for $\Delta RPQ$.
Sides $PR = 6\text{ cm}$ and $PQ = 4\text{ cm}$, included angle $\angle P = 60^{\circ}$.
\[ \frac{PR}{PQ} = \frac{6}{4} = 1.5 \]
Step 3: Ratio for $\Delta XZY$.
Sides $XZ = 9\text{ cm}$ and $YZ = 6\text{ cm}$, included angle $\angle Z = 60^{\circ}$.
\[ \frac{XZ}{YZ} = \frac{9}{6} = 1.5 \]
Step 4: Ratio for $\Delta LMN$.
Sides $LN = 3\text{ cm}$ and $MN = 4\text{ cm}$, included angle $\angle N = 60^{\circ}$.
\[ \frac{LN}{MN} = \frac{3}{4} = 0.75 \]
Step 5: Match the ratios.
All three included angles equal $60^{\circ}$, so the deciding factor is the side ratio. $\Delta RPQ$ gives $1.5$ and $\Delta XZY$ also gives $1.5$, an exact match. $\Delta LMN$ gives $0.75$, which does not match either of the other two, so it is not similar to them.
Final Answer:
Since the included angles are equal and the ratios of the sides forming them match only for $\Delta RPQ$ and $\Delta XZY$, these two triangles are similar. This is option (A).
\[ \boxed{\Delta RPQ \sim \Delta XZY} \]
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