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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 \(\underline{PR}\).

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Chapter 7

Let \(\triangle \mathrm{ABC}\) be a triangle such that $\angle \mathrm{B} \cong \angle \mathrm{C}$. Use the ASA Theorem to prove that \(\mathrm{AB}=\mathrm{AC}\). (Don't use the Isosceles Triangle Theorem.)

Chapter 7

Given \(\angle 1 \cong \angle 2 ; \angle 3 \cong \angle 4\). Prove \(\mathrm{RM}=\mathrm{RN}\).

Chapter 7

Prove that the altitudes drawn to the legs of an isosceles triangle are congruent.

Chapter 7

Given: \(\underline{A E}\) and \(\underline{B D}\) are straight lines intersecting at C. \(\underline{B C}\) \(\cong \underline{D C} ; \angle B \cong \angle D\). Prove: \(\Delta A B C \cong \triangle E D C\).

Chapter 7

Given $\triangle \mathrm{ABC}, \underline{\mathrm{AC}} \cong \underline{\mathrm{BC}}, \angle 1 \cong \angle 2 .\( Prove: \)\angle 3 \cong \angle 4$

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