Question

In the figure, the value of $\angle P$ will be: 

• A. $20^\circ$
• B. $40^\circ$
• C. $60^\circ$
• D. $80^\circ$

Hint:

When two triangles are similar, check corresponding sides to confirm the similarity ratio. Then match the corresponding angles directly.

Answer 1

The Correct Option is B

Step 1: Observe the two triangles
The two triangles $\triangle ABC$ and $\triangle PQR$ are given. Their sides are in proportion:
\[ AB : PQ = AC : PR = BC : QR \] Check: \[ \frac{AB}{PQ} = \frac{3.8}{7.6} = \frac{1}{2}, \frac{AC}{PR} = \frac{3\sqrt{3}}{6\sqrt{3}} = \frac{1}{2}, \frac{BC}{PQ} = \frac{6}{12} = \frac{1}{2} \] Hence, $\triangle ABC \sim \triangle PQR$ (by SSS similarity).

Step 2: Corresponding angles of similar triangles
$\triangle ABC \sim \triangle PQR$ means: \[ \angle A = \angle P, \angle B = \angle Q, \angle C = \angle R \]

Step 3: Value of $\angle P$
From $\triangle ABC$, $\angle A = 80^\circ$.
So, in $\triangle PQR$, $\angle P = 80^\circ$. Wait! Let's carefully check the figure again:
- In $\triangle ABC$, $\angle A = 80^\circ$, $\angle B = 60^\circ$.
- Then $\angle C = 180^\circ - (80^\circ+60^\circ) = 40^\circ$.
But from similarity, $\angle C = \angle R$, so $\angle A = \angle P$. Thus $\angle P = 80^\circ$. \[ \boxed{\angle P = 80^\circ} \] So correct option is (D) $80^\circ$.