Chapter 3.2, Problem 26PSC

To determine

To show that two triangles are congruent.

Answer to Problem 26PSC

  $\Delta ABCand\Delta AEDarecongruent.$

Explanation of Solution

Given information:

  $\begin{array}{l}\overline{AB} \cong \overline{AE};\hfill \\ \overrightarrow{AE}and\overrightarrow{AC}trisect\angle BAD,\hfill \\ \overline{AB}\perp \overline{BC},\hfill \\ \overline{AE}\perp \overline{DE}.\hfill \end{array}$

Formula used:

The below properties are used:

If an angle is trisected, it is divided into three congruent angles.

If two segments are perpendicular, they form a right angle.

Proof:

The below two statements are given:

  $\overline{AB} \cong \overline{AE}$

  $\overrightarrow{AE}and\overrightarrow{AC}trisect\angle BAD$

  

If an angle is trisected, it is divided into three congruent angles.

  $\angle 1\cong \angle 2\cong \angle 3$

By addition property,

  $\angle BAC \cong \angle DAE$

It is given,

  $\overline{AB}\perp \overline{BC}$

If two segments are perpendicular, they form a right angle.

  $\angle ABCisrightangle.$

It is given,

  $\overline{AE}\perp \overline{DE}$

If two segments are perpendicular, they form a right angle.

  $\angle AEDisrightangle.$

Right angles are congruent.

  $\angle ABC \cong \angle AED$

  $\therefore \Delta ABC\cong \Delta AED$