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Chapter 7: Chapter 7

Problem 106

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

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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 \(\underline{PR}\).
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TABLE OF CONTENTS :
TABLE OF CONTENTS

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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