Question

alex is writing statements to prove that the sum of the measures of interior angles of triangle pqr is equal to 180°. line m is parallel to line n. which is a true statement he could write? angle hrq measures 40°. angle pqr measures 60°. angle prq measures 80°. angle pqr measures 40°.

Explanation:

Step1: Use alternate - interior angles property

Since line \(n\) is parallel to line \(m\), alternate - interior angles are equal. The angle of \(40^{\circ}\) on line \(n\) and the angle at \(Q\) in \(\triangle PQR\) are alternate - interior angles, and the angle of \(60^{\circ}\) on line \(n\) and the angle at \(R\) in \(\triangle PQR\) are alternate - interior angles.

Step2: Recall angle - sum property of a triangle

The sum of interior angles of a triangle is \(180^{\circ}\). In \(\triangle PQR\), if one angle is \(40^{\circ}\) and another is \(60^{\circ}\), then the third angle \(\angle QPR=180-(40 + 60)=80^{\circ}\).

Step3: Analyze each option

• Option 1: \(\angle RHQ\) and the \(40^{\circ}\) angle are corresponding angles, so \(\angle RHQ = 40^{\circ}\), not \(45^{\circ}\).
• Option 2: \(\angle PCR\) and the \(60^{\circ}\) angle are corresponding angles, so \(\angle PCR=60^{\circ}\).
• Option 3: \(\angle PRQ\) is the angle at \(R\) of \(\triangle PQR\) which is \(60^{\circ}\), not \(80^{\circ}\).
• Option 4: \(\angle PQR\) is the angle at \(Q\) of \(\triangle PQR\) which is \(40^{\circ}\), not \(45^{\circ}\).

Answer:

Angle PCR measures \(60^{\circ}\)