Question

list the sides of the triangle in order from shortest to longest.

Explanation:

Step1: Find the measure of angle \( Q \)

The sum of angles in a triangle is \( 180^{\circ} \). Let \( \angle Q=x \). Then \( x + 87^{\circ}+46^{\circ}=180^{\circ} \). So \( x=180^{\circ}-(87^{\circ} + 46^{\circ})=47^{\circ} \).

Step2: Relate angles and sides

In a triangle, the larger the angle, the longer the side opposite it.

• \( \angle R = 46^{\circ} \), \( \angle Q=47^{\circ} \), \( \angle P = 87^{\circ} \)
• The side opposite \( \angle R \) is \( PQ \), the side opposite \( \angle Q \) is \( PR \), and the side opposite \( \angle P \) is \( QR \)

Since \( 46^{\circ}<47^{\circ}<87^{\circ} \), the order of sides from shortest to longest is based on the angles they are opposite.

Answer:

\( PQ, PR, QR \)