Given: M is the midpoint of LN . ∠LMP ≅ ∠NMO, ON⊥NM , PL⊥LM
Prove: ∆NMO ≅ ∆LMP
∠LMP ≅ ∠NMO (given)
LM ≅ NM (since M is midpoint)
∠L ≅ ∠N (given: both ⊥)
so, ASA --> ≅
To prove that ∆NMO ≅ ∆LMP, we can use several congruent triangles and angle relationships. Here's a step-by-step proof:
Step 1: Given that M is the midpoint of LN, we can establish the congruence of LM and MN because M is the midpoint of the side LN.
∆LPM ≅ ∆NPM by SAS congruence (Side-Angle-Side congruence)
Step 2: Given that ∠LMP ≅ ∠NMO and PL ⊥ LM, we can establish that ∆LPM and ∆NMO are similar.
∠LMP ≅ ∠NMO: Given
PL ⊥ LM: Given
Therefore, ∆LPM ∼ ∆NMO by AA similarity (Angle-Angle similarity)
Step 3: Given that ON ⊥ NM, we know that ∠ONM and ∠OMP are right angles. Hence, ∆NMO and ∆LPM are both right triangles.
ON ⊥ NM: Given
∠ONM and ∠OMP are right angles: Definition of perpendicular lines
Step 4: Using the congruence of ∆LPM ≅ ∆NPM established in step 1 and the similarity of ∆LPM ∼ ∆NMO established in step 2, we can conclude that ∆NMO ≅ ∆LPM.
∆LPM ≅ ∆NPM: From step 1 (SAS congruence)
∆LPM ∼ ∆NMO: From step 2 (AA similarity)
Since two triangles that are congruent are also similar, and two similar triangles are also congruent (by definition), we can conclude that ∆NMO ≅ ∆LPM.
Therefore, we have proven that ∆NMO ≅ ∆LPM.
To prove that ∆NMO ≅ ∆LMP, we will use the concept of triangle congruence. Specifically, we can use either the Side-Angle-Side (SAS) Congruence Postulate or the Angle-Side-Angle (ASA) Congruence Postulate.
Here's how we can prove it using the SAS Congruence Postulate:
1. Given that M is the midpoint of LN, we know that LM = MN (by the definition of a midpoint).
2. Given that ∠LMP ≅ ∠NMO, we have an angle pair that is congruent.
3. Given that ON⊥NM and PL⊥LM, we have two pairs of perpendicular lines.
Now, let's list our steps:
Step 1: Since M is the midpoint of LN, we have LM = MN.
Step 2: ∠LMP ≅ ∠NMO (given).
Step 3: ON ⊥ NM and PL ⊥ LM (given).
Step 4: Now, we can show that ∆NMO ≅ ∆LMP using the SAS Congruence Postulate.
4a. OS ⊥ ML (given).
4b. OS ⊥ NM (definition of perpendicular lines).
4c. Therefore, ∠MOS ≅ ∠OMS (by the definition of perpendicular lines).
4d. LN ≅ LN (reflexive property of congruence).
4e. ∆NOS ≅ ∆LOS (by the SAS Congruence Postulate).
4f. ∠NOS ≅ ∠LOS (corresponding parts of congruent triangles are congruent).
4g. ∠NMO ≅ ∠LMP (given).
4h. Therefore, ∆NMO ≅ ∆LMP (by the SAS Congruence Postulate).
And that completes the proof! We have shown that ∆NMO is congruent to ∆LMP using the SAS Congruence Postulate.