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Problem 106

# If $$\angle 3 \cong \angle 4$$ and $$Q \underline{M}$$ bisects $$\angle P Q R$$, prove that $$M$$ is the midpoint of $$\underline{P R}$$.

Expert verified
In summary, given that $$\angle 3 \cong \angle 4$$ and QM bisects $$\angle PQR$$, we proved that M is the midpoint of segment PR by showing $$\triangle QMP \cong \triangle QMR$$ using the AAS theorem ($$\angle 3 \cong \angle 4$$, $$\angle PQM \cong \angle QMR$$, and $$\overline{QM} \cong \overline{QM}$$). This congruence led to the conclusion $$\overline{MP}\cong\overline{MR}$$, confirming M as the midpoint of $$\underline{PR}$$.
See the step by step solution

## Step 1: Given angle congruence

We are given that angle 3 is congruent to angle 4, which means $$\angle 3 \cong \angle 4$$.

## Step 2: Congruence of angles PQM and QMR

Since QM bisects angle PQR, it divides angle PQR into two congruent angles by the definition of angle bisector. Therefore, $$\angle PQM \cong \angle QMR$$.

## Step 3: Congruence of side QM

In both triangles QMP and QMR, segment QM is common. By the reflexive property of congruence, we have $$\overline{QM} \cong \overline{QM}$$.

## Step 4: Proving triangle congruence by AAS theorem

We have $$\angle 3 \cong \angle 4$$, $$\angle PQM \cong \angle QMR$$, and $$\overline{QM} \cong \overline{QM}$$. By the Angle-Angle-Side theorem, triangles QMP and QMR are congruent. Therefore, $$\triangle QMP \cong \triangle QMR$$.

## Step 5: Proving M as the midpoint of PR

Since triangles QMP and QMR are congruent, their corresponding sides are also congruent. Therefore, $$\overline{MP} \cong \overline{MR}$$. Thus, M is the midpoint of segment PR as it divides the segment into two equal parts. Hence, we have proved that M is the midpoint of $$\underline{PR}$$.

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