Chapter 4.5, Problem 16WE

To determine

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

Answer to Problem 16WE

Explanation of Solution

Given information:

Given statement:

In two congruent triangles, if segments are drawn from two corresponding vertices perpendicular to the opposite sides, then those segments are congruent.

Calculation:

Consider that two triangles $\Delta ABC$ and $\Delta XYZ$ are congruent

Next step draw a perpendicular $\overline{AD}$ on side $\overline{BC}$ and $\overline{XP}$ on side $\overline{YZ}$

Next step write two columns prove

Given: $\Delta ABC\cong \Delta XYZ$

  $\begin{array}{l}\overline{AD}\perp \overline{BC}\\ \overline{XP}\perp \overline{YZ}\end{array}$

To prove: $\overline{AD}\cong \overline{XP}$

Proof:

The two columns proof is stated below:

Statements	Reasons
1. $\Delta ABC\cong \Delta XYZ$	Given
2. $\overline{AD}\perp \overline{BC}\text{and}\overline{XP}\perp \overline{YZ}$	Given
3. $\overline{AB}\cong \overline{XY}$	$\Delta ABC\cong \Delta XYZ$
4. $\angle B\cong \angle Y$	Corresponding angles of congruent triangles
5. $\angle ADP\cong \angle XPY$	$\overline{AD}\perp \overline{BC}\text{and}\overline{XP}\perp \overline{YZ}$
6. $\Delta ADB\cong \Delta XPY$	By AAS
7. $\overline{AD}\cong \overline{XP}$	Corresponding sides of congruent triangles