Given MN is perpendicular to MP, PQ is perpendicular to MP, and NR is congruent to OR. Prove triangle NRP is congruent to triangle QRM

Question

asked May 27, 2023 92.3k views

1 Answer

Prithvi Raj Nandiwal · Answer 1 · 2023-06-03T12:24:04+0000

1.
$\overline{MN} \perp \overline{MP}$ ,
$\overline{PQ} \perp \overline{MP}$ (given)

2.
$\angle NMP$ and
$\angle MPQ$ are right angles (definition of perpendicular lines)

3.
$\angle NMP \cong \angle MPQ$ (all right angles are congruent)

4.
$\overline{NR} \cong \overline{QR}$ (given)

5.
$\angle NRM \cong \angle QRP$ (vertical angles)

6.
$\triangle PRQ \cong \triangle MRN$ (AAS)

7.
$\overline{MR} \cong \overline{RP}$ (CPCTC)

8.
$\angle NRP \cong \angle MRQ$ (vertical angles)

9.
$\triangle NRP \cong \triangle QRM$ (SAS)