What are the missing reasons in the two-column proof?
Given: modifying above with right arrow upper M upper O bisects angle upper P upper M upper N and modifying above with right arrow upper O upper M bisects angle upper P upper O upper N
Prove: triangle upper P upper M upper O congruent to triangle upper N upper M upper O
statements | reasons
1. Mo bisects angle PMN 1.?
2. Angle PMO~=Angle NMO 2.?
3. MO ~=MO 3.?
4. OM bisects angle PON 4. ?
5. angle POM ~= angle NOM 5. ?
6. triangle PMO ~= triangle NMO 6.?
1. Mo bisects angle PMN 1. Given
2. Angle PMO~=Angle NMO 2. Vertical angles are congruent
3. MO ~=MO 3. Reflexive property of congruence
4. OM bisects angle PON 4. Given
5. angle POM ~= angle NOM 5. Vertical angles are congruent
6. triangle PMO ~= triangle NMO 6. SAS Congruence
To determine the missing reasons in the two-column proof, we need to use the information given in the "Given" statement and apply relevant geometric properties or theorems.
Here is how you can fill in the missing reasons:
1. Mo bisects angle PMN 1. Given
2. Angle PMO ~= Angle NMO 2. Vertical Angles are Congruent (Alternate Exterior Angles Theorem)
3. MO ~= MO 3. Reflexive Property of Congruence
4. OM bisects angle PON 4. Given
5. angle POM ~= angle NOM 5. Vertical Angles are Congruent (Alternate Interior Angles Theorem)
6. triangle PMO ~= triangle NMO 6. Angle-Angle-Side (AAS) Congruence Postulate
Therefore, the missing reasons are:
1. Given
2. Vertical Angles are Congruent (Alternate Exterior Angles Theorem)
3. Reflexive Property of Congruence
4. Given
5. Vertical Angles are Congruent (Alternate Interior Angles Theorem)
6. Angle-Angle-Side (AAS) Congruence Postulate
To determine the missing reasons in the two-column proof, we need to look for corresponding statements or facts that can be used to support each statement. Let's analyze each statement:
1. Mo bisects angle PMN: This statement is given in the prompt itself. The reason for this statement is provided, which is "Given."
2. Angle PMO ~= Angle NMO: This statement refers to the angles formed by the segments Mo and MN. To prove this statement, we can use the Angle Bisector Theorem, which states that if a line bisects an angle, it divides the angle into two congruent parts. The reason for this statement is "Angle Bisector Theorem."
3. MO ~= MO: This statement simply states that segment MO is congruent to itself. The reason for this statement is "Reflexive Property of Congruence" or "Identity Property of Congruence."
4. OM bisects angle PON: This statement is given in the prompt itself. The reason for this statement is provided, which is "Given."
5. Angle POM ~= Angle NOM: This statement refers to the angles formed by the segments OM and ON. To prove this statement, we can again use the Angle Bisector Theorem, since the segment OM bisects the angle PON. The reason for this statement is "Angle Bisector Theorem."
6. Triangle PMO ~= Triangle NMO: This statement refers to the congruence of the two triangles formed by the given information and the previous statements. To prove this statement, we can use the Angle-Angle-Side (AAS) Congruence Postulate, which states that if two angles and the included side of one triangle are congruent to the corresponding two angles and included side of another triangle, the triangles are congruent. The reason for this statement is "AAS Congruence Postulate."
By using these reasons in the two-column proof, each statement can be supported and the proof can be completed.