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
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Explanation:

Step1: Recall the triangle angle - side relationship

In a triangle, the larger side is opposite the larger angle, and the smaller side is opposite the smaller angle. This is a fundamental property of triangles.

Step2: Identify the sides and their opposite angles in \(\triangle OPQ\)

• The side \(PQ = 16\) is opposite angle \(O\) (since in \(\triangle OPQ\), side \(PQ\) is opposite vertex \(O\)).
• The side \(QO=15\) is opposite angle \(P\) (side \(QO\) is opposite vertex \(P\)).
• The side \(OP = 7\) is opposite angle \(Q\) (side \(OP\) is opposite vertex \(Q\)).

Step3: Compare the lengths of the sides

We have the side lengths: \(PQ = 16\), \(QO = 15\), \(OP=7\). So, \(PQ>QO > OP\).

Step4: Relate side lengths to angle measures

Since the larger side is opposite the larger angle:

• Since \(PQ>QO\), the angle opposite \(PQ\) (which is \(\angle O\)) is larger than the angle opposite \(QO\) (which is \(\angle P\)), so \(m\angle O>m\angle P\).
• Since \(QO > OP\), the angle opposite \(QO\) (which is \(\angle P\)) is larger than the angle opposite \(OP\) (which is \(\angle Q\)), so \(m\angle P>m\angle Q\).
• Combining these two inequalities, we get \(m\angle O>m\angle P > m\angle Q\).

Answer:

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