Question

If \(\Delta ABC ∼ \Delta PQR\),\(∠A=32°,∠R=65°\) then \(∠B=?\)

• A. \(93°\)
• B. \(83°\)
• C. \(73°\)
• D. \(63°\)

Answer 1

The Correct Option is B

To solve the problem, we need to determine the measure of \(\angle B\) in \(\triangle ABC\), given that \(\triangle ABC \sim \triangle PQR\), \(\angle A = 32^\circ\), and \(\angle R = 65^\circ\).

1. Understanding Similar Triangles:
Since \(\triangle ABC \sim \triangle PQR\), the corresponding angles of the two triangles are equal. This means: \[ \angle A = \angle P, \quad \angle B = \angle Q, \quad \angle C = \angle R \] Given \(\angle A = 32^\circ\) and \(\angle R = 65^\circ\), we know: \[ \angle C = \angle R = 65^\circ \]

2. Sum of Angles in a Triangle:
The sum of the interior angles in any triangle is \(180^\circ\). For \(\triangle ABC\), we have: \[ \angle A + \angle B + \angle C = 180^\circ \] Substituting the known values \(\angle A = 32^\circ\) and \(\angle C = 65^\circ\): \[ 32^\circ + \angle B + 65^\circ = 180^\circ \]

3. Solving for \(\angle B\):
Combine the known angle measures: \[ 32^\circ + 65^\circ = 97^\circ \] Thus: \[ 97^\circ + \angle B = 180^\circ \] Solving for \(\angle B\): \[ \angle B = 180^\circ - 97^\circ = 83^\circ \]

Final Answer:
The measure of \(\angle B\) is \({83^\circ}\).