Chapter 4.5, Problem 13CE

To determine

To find: congruent criteria used to prove that the triangles are congruent.

Answer to Problem 13CE

Triangle are congruent by AAS congruence criteria

  $\Delta BAD,\Delta CAE$ is a pair of overlapping triangles.

Explanation of Solution

Given information: $\angle ABC\cong \angle ACB$ , $\overline{AE}\perp \overline{EC}$ and $\overline{AD}\perp \overline{DB}$

  

Consider $\Delta BAD,\Delta CAE$

As $\overline{AD}\perp \overline{DB}$ , $\angle BDA=90°$

As $\overline{AE}\perp \overline{EC}$ , $\angle CEA=90°$

So,

  $\angle CEA\cong \angle BDA$ (i) (right angles are congruent to each other)

Also,

  $\angle BAD\cong \angle CAE$ (ii) (common angles in $\Delta BAD,\Delta CAE$ )

As $\angle ABC\cong \angle ACB$ ,

  $\overline{AB}\cong \overline{AC}$ (iii)

(In a triangle, sides opposite to equal angles are equal)

From (i), (ii) and (iii),

  $\Delta BAD\cong \Delta CAE$ (by AAS congruence criteria)