Question

1. △hmp is isosceles triangle with base (overline{mp}). also, (overline{np}perpoverline{om}) and (overline{nm}congoverline{op}). name the postulate or theorem you can use to prove (△nhmcong△ohp).

aas theorem
cpctc
hl theorem
asa postulate

Explanation:

Step1: Analyze the given information

We know that \(\overline{NP}\perp\overline{OM}\), so \(\angle NHM=\angle OHP = 90^{\circ}\) (right angles). Also, \(\overline{NM}\cong\overline{OP}\) (given). Since \(\triangle HMP\) is isosceles with base \(\overline{MP}\), then \(\angle HMP=\angle HPM\), and by the vertical - angle theorem, \(\angle NHM\) and \(\angle OHP\) are right angles, and \(\angle MHN=\angle PHO\) (vertical angles). So, we have two angles and a non - included side.

Step2: Recall triangle congruence theorems

• AAS (Angle - Angle - Side) Theorem: If two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.
• CPCTC (Corresponding Parts of Congruent Triangles are Congruent): This is used to prove parts of triangles are congruent after the triangles are already proven congruent, not to prove triangle congruence.
• HL (Hypotenuse - Leg) Theorem: Requires a right triangle, hypotenuse and a leg. Here we don't have information about hypotenuse and leg in the sense of \(HL\) for these two triangles \(\triangle NHM\) and \(\triangle OHP\).
• ASA (Angle - Side - Angle) Theorem: Requires two angles and the included side.

Since we have two angles (\(\angle NHM=\angle OHP\) and \(\angle MNH=\angle POH\) (because \(\angle HMP=\angle HPM\) and using angle - sum property in triangles) and a non - included side (\(\overline{NM}\cong\overline{OP}\)), we use the AAS Theorem.

Answer:

A. AAS Theorem