Chapter 4.7, Problem 21WE

(a)

To determine

Prove $\overline{BD}\cong \overline{CE}$ .

Explanation of Solution

Given:

  $\overline{AB}\cong \overline{AC}$

  $\overline{BD}\perp \overline{AC}$

  $\overline{CE}\perp \overline{AB}$

Calculation:

The diagram is :

  

Consider the $\Delta BAD\text{ and }\Delta \text{CAE}$

  $\overline{BD}\perp \overline{AC}$

  $\overline{CE}\perp \overline{AB}$

So, $\angle ADB\cong \angle \text{AEC}$

  $\angle BAD\cong \angle CAE$ .....[Reflexive Property]

  $\overline{AB}\cong \overline{AC}$

So, by AAS Postulate , $\Delta BAD\cong \Delta \text{CAE}$ .

Hence, $\overline{BD}\cong \overline{CE}$

Hence proved.

(a)

To determine

The result in part (a) can be stated as a theorem about certain atltitudes.State this theorem in your own words.

Answer to Problem 21WE

In an isosceles triangle , the altitudes from the base angle are congruent.

Explanation of Solution

Given:

From part (a) , we have :

  $\overline{AB}\cong \overline{AC}$

  $\overline{BD}\perp \overline{AC}$

  $\overline{CE}\perp \overline{AB}$

Then , $\overline{BD}\cong \overline{CE}$

Calculation:

The diagram is :

  

Since $\overline{AB}\cong \overline{AC}$ , so the triangle ABC is isosceles.

  $\overline{BD}\perp \overline{AC}$

  $\overline{CE}\perp \overline{AB}$

So, BD and CE are altitudes from the base angles.

Then , we proved in part (a), $\overline{BD}\cong \overline{CE}$

So, we can state that :

In an isosceles triangle , the altitudes from the base angle are congruent.