Given: MO bisects 4PMN and OM bisects 4PON
Prove: 4PMO =ANMO
1. MO bisects <PMN
2. LPMO = <NMO
3. МО = МО
4. OM bisects PON
5. LPOM = LNOM
6. PMO = ANMO
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It seems that there is a typo in the prove statement: "Prove: ∠PMO = ANMO" does not make sense because ∠PMO is an angle and ANMO is not defined in this context. I am going to assume that you meant to prove that triangle PMO is congruent to triangle NMO, based on the information given. If that's the case, here is how the proof would be structured:
1. Given: MO bisects ∠PMN.
2. ∠PMO = ∠NMO (Definition of angle bisector)
3. MO = MO (Reflexive property of congruence)
4. Given: OM bisects ∠PON.
5. ∠POM = ∠NOM (Definition of angle bisector)
6. Triangle PMO ≅ Triangle NMO (ASA Postulate)
Let's fill in the corresponding reasons for each step:
1. Given
2. Definition of an angle bisector
3. Reflexive Property of Equality
4. Given
5. Definition of an angle bisector
6. Angle-Side-Angle (ASA) Congruence Postulate
Explanation of the steps:
1-2, 4-5: The given information states that MO bisects ∠PMN and OM bisects ∠PON, so by definition, ∠PMO = ∠NMO and ∠POM = ∠NOM since an angle bisector divides an angle into two congruent angles.
3: The Reflexive Property of Equality simply states that any geometric figure is congruent to itself, which is obviously true for segment MO.
6: Using the Angle-Side-Angle (ASA) Congruence Postulate, we can conclude that ∆PMO ≅ ∆NMO because we have two angles and the included side between them that are corresponding and congruent in both triangles.