Question

the triangle pqr is rotated counterclockwise about the origin to form a triangle pqr.
which statement is true?
○ the measure of ∠pqr is 51°.
○ the measure of ∠pqr is 83°.
○ the measure of ∠qpr is 83°.
○ the measure of ∠qrp is 46°.

Explanation:

Rotation is a rigid transformation, meaning all side lengths and angle measures of the triangle remain unchanged after rotation. In the original triangle \(PQR\), \(\angle PQR = 83^\circ\), which corresponds to \(\angle P'Q'R'\) in the rotated triangle \(P'Q'R'\). We verify each option:

1. The measure of \(\angle P'Q'R'\) is not \(51^\circ\) (this is \(\angle PRQ\) in the original triangle).
2. The measure of \(\angle P'Q'R'\) matches the original \(\angle PQR = 83^\circ\), so this is true.
3. The measure of \(\angle Q'P'R'\) should be \(46^\circ\) (matching original \(\angle QPR\)), not \(83^\circ\).
4. The measure of \(\angle Q'R'P'\) should be \(51^\circ\) (matching original \(\angle QRP\)), not \(46^\circ\).

Answer:

The measure of \(\angle P'Q'R'\) is \(83^\circ\).