Question

in the diagram, mq = qp = po = on. if np is greater than mp, which must be true?

segment op is longer than segment mq.
segment mn has the same length as segment mp.
the measure of angle q equals the measure of angle o.
angle o is larger than angle q.

Explanation:

Step1: Analyze given equal segments

Given \(MQ = QP = PO = ON\), so \(MQ = OP\) (both equal to \(QP\)). Thus, option 1 is false.

Step2: Relate sides to angles (Triangle Law)

In \(\triangle MQP\) and \(\triangle NPO\):

• \(MQ = PO\), \(QP = ON\) (given equal segments).
• \(NP > MP\) (given). By the converse of the triangle angle-side relationship: in two triangles with two pairs of equal sides, the triangle with the longer third side has the larger included angle.

The included angle for \(MP\) in \(\triangle MQP\) is \(\angle Q\); for \(NP\) in \(\triangle NPO\) is \(\angle O\). Since \(NP > MP\), \(\angle O > \angle Q\).

Answer:

D. Angle O is larger than angle Q.