Question

question in \\( \triangle \mathrm { pqr }, \mathrm { m } \angle \mathrm { p } = 89 ^ { \circ } \\) and \\( \mathrm { m } \angle \mathrm { q } = 31 ^ { \circ } \\). which list has the sides of \\( \triangle \mathrm { pqr } \\) in order from longest to shortest? answer \\( \bigcirc \mathrm { pq }, \mathrm { rp }, \mathrm { qr } \\) \\( \bigcirc \mathrm { rp }, \mathrm { qr }, \mathrm { pq } \\) \\( \bigcirc \mathrm { rp }, \mathrm { pq }, \mathrm { qr } \\) \\( \bigcirc \mathrm { pq }, \mathrm { qr }, \mathrm { rp } \\) \\( \bigcirc \mathrm { qr }, \mathrm { pq }, \mathrm { rp } \\) \\( \bigcirc \mathrm { qr }, \mathrm { rp }, \mathrm { pq } \\)

Explanation:

Step1: Find the measure of ∠R

In a triangle, the sum of interior angles is \(180^{\circ}\). So, \(m\angle R=180^{\circ}-m\angle P - m\angle Q\). Substitute \(m\angle P = 89^{\circ}\) and \(m\angle Q = 31^{\circ}\):
\(m\angle R=180 - 89 - 31=60^{\circ}\)

Step2: Order the angles from largest to smallest

Compare the angle measures: \(m\angle P = 89^{\circ}\), \(m\angle R = 60^{\circ}\), \(m\angle Q = 31^{\circ}\). So, \(m\angle P>m\angle R>m\angle Q\)

Step3: Relate angles to their opposite sides

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

• Opposite \(m\angle P\) ( \(89^{\circ}\)) is side \(QR\)
• Opposite \(m\angle R\) ( \(60^{\circ}\)) is side \(PQ\)
• Opposite \(m\angle Q\) ( \(31^{\circ}\)) is side \(RP\)

So, the order of sides from longest to shortest (since larger angle → longer opposite side) is: \(QR\) (opposite \(89^{\circ}\)), \(PQ\) (opposite \(60^{\circ}\)), \(RP\) (opposite \(31^{\circ}\))

Answer:

QR, PQ, RP