Chapter 4.5, Problem 17WE

To determine

To draw: and label a diagram, and to prove the given statement.

Explanation of Solution

Given information:

Given statement:

Segments are drawn from the endpoints of the base of an isosceles triangle perpendicular to the opposite legs, and then those segments are congruent.

Calculation:

Suppose if segments are drawn from the endpoints of the base of an isosceles triangle perpendicular to the opposite legs, then prove those segment are congruent.

The provided data is as follows:

Here, $\Delta XYZ$ is isosceles with $\overline{XY}\cong \overline{XZ}$

Also, it is provided that

  $\begin{array}{l}\overline{ZA}\perp \overline{XY}\\ \overline{YB}\perp \overline{XZ}\end{array}$

To prove: $\overline{ZA}\cong \overline{YB}$

Show that $\Delta XBY\cong \Delta XAZ$

Consider these two triangles and follow the following steps provided in the table below:

Statements	Reasons
1. $\overline{ZA}\perp \overline{XY};\overline{YB}\perp \overline{XZ}$	Given
2. $m\angle XBY=90°;m\angle XAZ=90°$	Definition of perpendicular line
3. $\angle XBY\cong \angle XAZ$	Definition of congruent angle
4. $\angle X\cong \angle X$	By reflexive property
5. $\overline{XY}\cong \overline{XZ}$	Provided
6. $\Delta XBY\cong \Delta XAZ$	AAS theorem
7. $\overline{ZA}\cong \overline{YB}$	CPCT

Therefore, the triangles $\Delta XBY$ and $\Delta XAZ$ are congruent by AAS property.

This implies that by congruent parts of congruent triangles CPCT are equal, that is $\overline{ZA}\cong \overline{YB}$ .