Question

Given: ∠ONL ≅ ∠MNL, ∠O and ∠M are right angles.
Prove: Line LM ≅ Line NO
Statements:
1)∠ONL≅ ∠MNL, ∠O and ∠M are right angles
2)∠O≅ ∠M
3)Line LN ≅ Line LN
4)Triangle ONL ≅ Triangle MLN
5)Line M ≅ Line NO

Reasons:
1)Given
2)All right angles are congruent
3)?
4)?
5)?

Answer 1

To prove that line LM is congruent to line NO, we can use the SAS (Side-Angle-Side) congruence theorem.

Here is the proof:

1) Given: ∠ONL ≅ ∠MNL, ∠O and ∠M are right angles
2) All right angles are congruent
3) ∠O ≅ ∠M (given angles are congruent)
4) Line ON ≅ Line LN (given ∠ONL ≅ ∠MNL)
5) ∠NO ≅ ∠NM (corresponding angles are congruent)
6) Triangle ONL ≅ Triangle MLN (by SAS congruence: sides ON ≅ ML (reflexive property), ∠NO ≅ ∠NM, and ∠ONL ≅ ∠MNL)
7) Line LM ≅ Line NO (by CPCTC - Corresponding Parts of Congruent Triangles are Congruent, so LN ≅ LN and ∠NLO ≅ ∠NML)
8) Thus, line LM is congruent to line NO.

The reasons for each step are provided in parentheses.