Question

angle side relationship
question
in \\( \triangle \mathrm { pqr } \\), \\( \mathrm { qr } = 15 \\), \\( \mathrm { rp } = 16 \\), and \\( \mathrm { pq } = 18 \\). which statement about the angles of \\( \triangle \mathrm { pqr } \\) must be true?
answer
\\( \mathrm { m } \angle p > \mathrm { m } \angle r > \mathrm { m } \angle q \\)
\\( \mathrm { m } \angle p > \mathrm { m } \angle q > \mathrm { m } \angle r \\)
\\( \mathrm { m } \angle r > \mathrm { m } \angle q > \mathrm { m } \angle p \\)
\\( \mathrm { m } \angle r > \mathrm { m } \angle p > \mathrm { m } \angle q \\)
\\( \mathrm { m } \angle q > \mathrm { m } \angle r > \mathrm { m } \angle p \\)
\\( \mathrm { m } \angle q > \mathrm { m } \angle p > \mathrm { m } \angle r \\)

Explanation:

Step1: Recall the Angle-Side Relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. So we first identify the lengths of the sides and their opposite angles.
In $\triangle PQR$:

• Side $QR = 15$ is opposite angle $P$ (since in $\triangle PQR$, side opposite $\angle P$ is $QR$).
• Side $RP = 16$ is opposite angle $Q$ (side opposite $\angle Q$ is $RP$).
• Side $PQ = 18$ is opposite angle $R$ (side opposite $\angle R$ is $PQ$).

Step2: Order the Sides by Length

Order the side lengths from largest to smallest: $PQ = 18 > RP = 16 > QR = 15$.

Step3: Relate Side Lengths to Angles

Since the larger side is opposite the larger angle, the angle opposite the largest side ($PQ = 18$) is the largest angle, the angle opposite the middle - length side ($RP = 16$) is the middle - sized angle, and the angle opposite the smallest side ($QR = 15$) is the smallest angle.

• The largest side $PQ = 18$ is opposite $\angle R$, so $m\angle R$ is the largest angle.
• The middle - length side $RP = 16$ is opposite $\angle Q$, so $m\angle Q$ is the middle - sized angle.
• The smallest side $QR = 15$ is opposite $\angle P$, so $m\angle P$ is the smallest angle.

So the order of the angles from largest to smallest is $m\angle R>m\angle Q > m\angle P$.

Answer:

m∠R > m∠Q > m∠P (corresponding to the option "m∠R > m∠Q > m∠P")