Question

Assertion (A) : \(\triangle ABC \sim \triangle PQR\) such that \(\angle A = 65^\circ\), \(\angle C = 60^\circ\). Hence \(\angle Q = 55^\circ\).
Reason (R) : Sum of all angles of a triangle is \(180^\circ\).

• A. Both Assertion (A) and Reason (R) are true and Reason (R) is correct explanation of Assertion (A)
• B. Both Assertion (A) and Reason (R) are true, but Reason (R) is not correct explanation for Assertion (A)
• C. Assertion (A) is true, but Reason (R) is false. (D) Assertion (A) is false, but Reason (R) is true

Answer 1

The Correct Option is A

Given:
- \(\triangle ABC \sim \triangle PQR\) (triangles are similar).
- \(\angle A = 65^\circ\), \(\angle C = 60^\circ\).
- Reason (R): Sum of all angles of a triangle is \(180^\circ\).

Step 1: Use the property of angles in \(\triangle ABC\)
Sum of angles in a triangle is \(180^\circ\), so:
\[ \angle A + \angle B + \angle C = 180^\circ \] Substitute known angles:
\[ 65^\circ + \angle B + 60^\circ = 180^\circ \] \[ \angle B = 180^\circ - (65^\circ + 60^\circ) = 180^\circ - 125^\circ = 55^\circ \]

Step 2: Use property of similar triangles
Since \(\triangle ABC \sim \triangle PQR\), corresponding angles are equal:
\[ \angle B = \angle Q \] So, \[ \angle Q = 55^\circ \]

Step 3: Conclusion on Assertion (A) and Reason (R)
- Assertion (A) states: Hence \(\angle Q = 55^\circ\) (True).
- Reason (R) states: Sum of angles in a triangle is \(180^\circ\) (True).
- Reason (R) is used to find \(\angle B\), which leads to finding \(\angle Q\).

Final Answer:
\[ \boxed{ \text{Both Assertion (A) and Reason (R) are true, and Reason (R) correctly explains Assertion (A). (Option A)} } \]