Question

Given OP bisects angle MON, PM is perpendicular to OM and PN is perpendicular to ON. Prove PN is congruent to PM

Answer 1

1.
`\overline{OP}` bisects
`\angle MON` (given)

2.
`\angle MOP \cong \angle PON` (definition of angle bisector)

3.
`\overline{PN} \perp \overline{ON}`,
`\overline{PM} \perp \overline{OM}` (given)

4.
`\angle PNO` and
`\angle PMO` are right angles (definition of perpendicular lines)

5.
`\angle PNO \cong \angle PMO` (all right angles are congruent)

6.
`\overline{OP} \cong \overline{OP}` (reflexive property)

7.
`\triangle MOP \cong \triangle NOP` (AAS)

8.
`\overline{PN} \cong \overline{PM}` (CPCTC)