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
A two column proof and a diagram are shown.
no columns. Why don't you make the table, and forget all that arrow stuff, and just name the segments or rays, as in MO bisects <PMN.
Give reasons where you have them, and take a stab at the missing ones.
To determine the missing reasons in the two-column proof, you need to analyze the information given in the given and prove statements, as well as the diagram provided.
First, let's review the information given:
Given:
1. $\overleftrightarrow{MO}$ bisects $\angle PMN$
2. $\overleftrightarrow{OM}$ bisects $\angle PON$
Prove:
$\triangle PMO$ ≅ $\triangle NMO$
Now, let's examine the steps in the two-column proof and identify the missing reasons:
Statement | Reason
--- | ---
1. $\overleftrightarrow{MO}$ bisects $\angle PMN$ | Given
2. $\angle PMO ≅ \angle MNO$ | Definition of Angle Bisector
3. $\overline{MO} ≅ \overline{MO}$ | Reflexive Property of Congruence
4. $\triangle PMO$ ≅ $\triangle NMO$ | ??
Based on the given information and the missing reasons, we can determine that Step 4 of the proof requires the reason "ASA congruence" since we have established congruent angles ($\angle PMO ≅ \angle MNO$) and congruent sides ($\overline{MO} ≅ \overline{MO}$).
Therefore, the missing reason in Step 4 of the two-column proof is "ASA congruence."