Chapter 4.2, Problem 22WE

To determine

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

Answer to Problem 22WE

Explanation of Solution

Given information:

Given statement:

In an isosceles triangle, if the angle between the congruent sides is bisected, then two congruent triangles are formed.

Calculation:

SSS Postulate: if there sides of triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

SAS Postulate: If two sides and the angle included between them of triangle are equal to the two corresponding sides and the angle included between them of another triangle, then the triangles are congruent.

ASA Postulate: If two angles and the included side of triangle are equal to the two corresponding angles and the included sides of another triangle, then the triangles are congruent.

Given:

An isosceles triangle $ABC$ such that $\overline{AC}\cong \overline{AB}$ ;

$AD$ is the bisector of $\angle BAC$

Prove: $\Delta ADC\cong \Delta ADB$

Proof:

Statements	Reasons
1. $AD$ is the bisector of   $\angle BAC$	Given
2. $\angle CAD\cong \angle BAD$	Definition of bisector of an angle
3. $\overline{AC}\cong \overline{AB}$	Given
4. $\overline{AD}\cong \overline{AD}$	Reflexive property
5. $\Delta ADC\cong \Delta ADB$	SAS Postulate