Chapter 4.6, Problem 24PPS

To determine

To prove:

  $\overline{WZ}\cong \overline{QY}$ for the given figure in the form of the paragraph

Explanation of Solution

Given information:

  

Proof:

Know that $\overline{XY}\cong \overline{XZ}$ and $W$ is midpoint of $\overline{XY}$ . $Q$ is midpoint of $\overline{XZ}$ . By the definition of the midpoint bisector $\overline{XW}\cong \overline{WY}$ since $W$ is midpoint of $\overline{XY}$ . Similarly $\overline{XQ}\cong \overline{QZ}$ . By using the segment addition postulates get that $\overline{XW}+\overline{WY}=\overline{XY}$ and $\overline{XQ}+\overline{QZ}=\overline{XZ}$ . On substitution:

  $\begin{array}{l}\overline{XW}+\overline{WY}=\overline{XQ}+\overline{QZ}\\ \overline{WY}+\overline{WY}=\overline{QZ}+\overline{QZ}\\ 2\overline{WY}=2\overline{QZ}\\ \overline{WY}=\overline{QZ}\end{array}$

According to the isosceles triangle theorem $\angle XYZ\cong \angle XZY$ . By the reflexive property $\overline{YZ}\cong \overline{ZY}$ . Use the side angle side theorem $\Delta WYZ\cong \Delta QZY$ . Know that corresponding parts of congruent triangles are congruent so, $\overline{WZ}\cong \overline{QY}$

Hence proved