Question

If \(\triangle ABC\) and \(\triangle PQR\) are similar triangles in which \(\angle A=47^\circ\) and \(\angle Q=83^\circ\), then \(\angle C\) is

• A. \(60^\circ\)
• B. \(70^\circ\)
• C. \(90^\circ\)
• D. \(50^\circ\)

Hint:

For similar triangles written as \(\triangle ABC \sim \triangle PQR\), match angles in order: \(A\leftrightarrow P\), \(B\leftrightarrow Q\), and \(C\leftrightarrow R\).

Answer 1

The Correct Option is D

Concept:
If two triangles are similar, then their corresponding angles are equal. For similar triangles: \[ \triangle ABC \sim \triangle PQR \] the corresponding angles are: \[ \angle A=\angle P \] \[ \angle B=\angle Q \] \[ \angle C=\angle R \] Also, the sum of angles of a triangle is: \[ 180^\circ \]

Step 1: Use the similarity relation.
Given: \[ \triangle ABC \sim \triangle PQR \] So, corresponding angles are equal: \[ \angle B=\angle Q \] Given: \[ \angle Q=83^\circ \] Therefore: \[ \angle B=83^\circ \]

Step 2: Use the angle sum property of triangle \(ABC\).
In \(\triangle ABC\): \[ \angle A+\angle B+\angle C=180^\circ \] Given: \[ \angle A=47^\circ \] and: \[ \angle B=83^\circ \] So: \[ 47^\circ+83^\circ+\angle C=180^\circ \] \[ 130^\circ+\angle C=180^\circ \] \[ \angle C=180^\circ-130^\circ \] \[ \angle C=50^\circ \] Hence, the correct answer is: \[ \boxed{(D)\ 50^\circ} \]