Chapter 4.2, Problem 23WE

To determine

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

Explanation of Solution

Given information:

Given statement:

In an isosceles triangle, if a segment is drawn from the vertex of the angle between the congruent sides to the mispoint of the opposite side, then 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 $\overline{AC}\cong \overline{AB}$ ;

$D$ is the midpoint of $\overline{CB}$

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

Proof:

Statements	Reasons
1. $D$ is the midpoint of   $\overline{CB}$	Given
2. $\overline{CD\cong \overline{BD}}$	Definition of midpoint
3. $\overline{AC}\cong \overline{AB}$	Given
4. $\overline{AD}\cong \overline{AD}$	Reflexive property
5. $\Delta ADC\cong \Delta ADB$	SAS Postulate