Question

question
in δopq, pq = 16, qo = 15, and op = 7. which statement about the angles of δopq must be true?
answer
○ m∠q > m∠p > m∠o ○ m∠p > m∠o > m∠q
○ m∠o > m∠q > m∠p ○ m∠o > m∠p > m∠q
○ m∠p > m∠q > m∠o ○ m∠q > m∠o > m∠p

Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger the length of a side, the larger the measure of the angle opposite that side. This is a fundamental property of triangles. So we need to first identify the lengths of the sides of \(\triangle OPQ\) and then determine the angles opposite each side.

The sides of \(\triangle OPQ\) are: \(PQ = 16\), \(QO=15\), and \(OP = 7\).

Let's identify the angles opposite each side:

• The side opposite \(\angle O\) is \(PQ\). So the length of the side opposite \(\angle O\) is \(PQ = 16\).
• The side opposite \(\angle P\) is \(QO\). So the length of the side opposite \(\angle P\) is \(QO = 15\).
• The side opposite \(\angle Q\) is \(OP\). So the length of the side opposite \(\angle Q\) is \(OP=7\).

Step2: Compare the side lengths

We compare the lengths of the sides: \(16>15 > 7\), that is \(PQ>QO > OP\).

Step3: Relate side lengths to angle measures

Since the angle opposite a longer side is larger, we have:

• Since \(PQ\) (opposite \(\angle O\)) is the longest side, \(m\angle O\) is the largest angle.
• Since \(QO\) (opposite \(\angle P\)) is the second - longest side, \(m\angle P\) is the second - largest angle.
• Since \(OP\) (opposite \(\angle Q\)) is the shortest side, \(m\angle Q\) is the smallest angle.

So the order of the angle measures from largest to smallest is \(m\angle O>m\angle P > m\angle Q\).

Answer:

\(m\angle O>m\angle P > m\angle Q\) (the option with this statement)