Question

In the figure, in $\triangle MNL$ and $\triangle PQR$, $\angle M = \angle Q = 70^\circ$, $MN = 3$ cm, $ML = 4.5$ cm, $PQ = 2$ cm, and $QR = 3$ cm. Then, the following correct relation will be: 

• A. $\triangle NML \sim \triangle QPR$
• B. $\triangle NML \sim \triangle QRP$
• C. $\triangle NML \sim \triangle PQR$
• D. None of these

Hint:

When two triangles have one equal angle and the sides including these angles are in proportion, they are similar by the SAS criterion.

Answer 1

The Correct Option is A

Step 1: Given data. 
For $\triangle MNL$: $MN = 3$ cm, $ML = 4.5$ cm, and $\angle M = 70^\circ$. 
For $\triangle PQR$: $PQ = 2$ cm, $QR = 3$ cm, and $\angle Q = 70^\circ$. 
Step 2: Compare sides including the equal angles. 
\[ \dfrac{MN}{QR} = \dfrac{3}{3} = 1, \quad \dfrac{ML}{PQ} = \dfrac{4.5}{2} = 2.25 \] These are not equal, but let’s check other possible corresponding sides. If we consider $\triangle NML$ and $\triangle QPR$: \[ \dfrac{MN}{QP} = \dfrac{3}{2} = 1.5, \quad \dfrac{ML}{QR} = \dfrac{4.5}{3} = 1.5 \] The sides are in the same ratio and included angles are equal (\(70^\circ\)). 
Step 3: Conclusion. 
Hence, by the SAS similarity criterion, \[ \triangle NML \sim \triangle QPR \]