Question

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?

• A. $\Delta RPQ$ and $\Delta XZY$
• B. $\Delta RPQ$ and $\Delta MNL$
• C. $\Delta XZY$ and $\Delta MNL$
• D. $\Delta RPQ$, $\Delta XZY$ and $\Delta MNL$ are similar to one another

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).

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given three triangles: $\Delta RPQ$, $\Delta XZY$, and $\Delta LMN$ (labeled with vertices $L, M, N$).
The lengths of two adjacent sides and the measure of their included angle are given for each triangle. We need to identify which pair of triangles are similar.

Step 2: Key Formula or Approach:
We will use the Side-Angle-Side (SAS) similarity criterion.
According to the SAS similarity criterion:
If one angle of a triangle is equal to one angle of another triangle, and the sides including these angles are proportional, then the two triangles are similar.

Step 3: Detailed Explanation:

• 1. Let us examine the parameters of $\Delta RPQ$:
- Adjacent sides to the given angle: $PR = 6\text{ cm}$ and $PQ = 4\text{ cm}$.
- Included angle: $\angle P = 60^\circ$.
- Ratio of these two sides (longer to shorter):
\[ \frac{PR}{PQ} = \frac{6}{4} = \frac{3}{2} = 1.5 \]

• 2. Let us examine the parameters of $\Delta XZY$:
- Adjacent sides to the given angle: $XZ = 9\text{ cm}$ and $YZ = 6\text{ cm}$.
- Included angle: $\angle Z = 60^\circ$.
- Ratio of these two sides (longer to shorter):
\[ \frac{XZ}{YZ} = \frac{9}{6} = \frac{3}{2} = 1.5 \]

• 3. Comparing $\Delta RPQ$ and $\Delta XZY$:
- The included angles are equal: $\angle P = \angle Z = 60^\circ$.
- The ratios of the corresponding sides are proportional:
\[ \frac{PR}{XZ} = \frac{6}{9} = \frac{2}{3} \]
\[ \frac{PQ}{YZ} = \frac{4}{6} = \frac{2}{3} \]
- Therefore, by the SAS similarity criterion:
\[ \Delta RPQ \sim \Delta XZY \]

• 4. Let us examine the third triangle $\Delta LMN$ (with vertices $L, M, N$):
- Adjacent sides to the given angle: $LN = 3\text{ cm}$ and $MN = 4\text{ cm}$.
- Included angle: $\angle N = 60^\circ$.
- Ratio of these sides:
\[ \frac{LN}{MN} = \frac{3}{4} = 0.75 \]
- Since the ratio $0.75$ does not match $\frac{3}{2}$ or its reciprocal, this triangle is not similar to the other two.


Step 4: Final Answer:
Only the triangles $\Delta RPQ$ and $\Delta XZY$ are similar, which corresponds to option (A).