Chapter 4.4, Problem 20WE

To determine

To prove: $\Delta ABC$ is isosceles.

Explanation of Solution

Given information:

Given figure,

Given,

  $\angle 4\cong \angle 7;\angle 1\cong \angle 3$

Calculation:

The Isosceles Triangle Theorem: If two sides of a triangle are congruent, then the angles opposite to those sides are congruent.

Given: $\angle 4\cong \angle 7;\angle 1\cong \angle 3$

Proof:

Statements	Reasons
1. $\angle 4\cong \angle 7$	Given
2. $\angle 5\cong \angle 6$	Angles adjacent to   $\cong$ angles are   $\cong$
3. $\overline{AL}\parallel \overline{AM}$	Converse of isosceles theorem
4. $\angle 1\cong \angle 3$	Given
5. $\Delta ABL\cong \Delta ACM$	ASA postulate
6. $\overline{AB}\cong \overline{AC}$	Corresponding parts   $\cong$
8. $\overline{AL}\parallel \overline{AM}$	Converse of isosceles theorem
9. $\angle 1\cong \angle 3$	Given
10. $\Delta ABL\cong \Delta ACM$	ASA postulate
11. $\overline{AB}\cong \overline{AC}$	Corresponding parts   $\cong$ triangles are   $\cong$
12. $\Delta ABC$ is isosceles	Definition of isosceles triangle