Question

1. what additional information is needed to prove ( \triangle pmocong\triangle pno ) by the hl theorem? ( overline{om}congoverline{on} ) ( overline{pm}congoverline{mp} ) ( angle mopcongangle nop ) no additional information is needed.

Explanation:

Step1: Recall the HL (Hypotenuse - Leg) Theorem

The HL Theorem states that if the hypotenuse and a leg of one right - triangle are congruent to the hypotenuse and a leg of another right - triangle, then the two right - triangles are congruent. In \(\triangle PMO\) and \(\triangle PNO\), \(\angle PMO=\angle PNO = 90^{\circ}\) (given right angles), and \(OP = OP\) (common hypotenuse).

Step2: Identify the legs

We need to show that the legs of the right - triangles are congruent. The legs are \(OM\) and \(ON\). If \(\overline{OM}\cong\overline{ON}\), then by the HL Theorem, \(\triangle PMO\cong\triangle PNO\).

• For \(\overline{PM}\cong\overline{MP}\), this is a reflexive property (\(\overline{PM}=\overline{PM}\)), but it doesn't help in the HL context for the two different triangles \(\triangle PMO\) and \(\triangle PNO\).
• For \(\angle MOP\cong\angle NOP\), this is an angle - related condition, and the HL Theorem is for side - side (hypotenuse - leg) in right - triangles, not for angles.

Answer:

\(\overline{OM}\cong\overline{ON}\)