Chapter 4.4, Problem 12CE

To determine

To explain: the corollary an equilateral triangle is also equiangular the two angles are congruent, then the sides opposite to those angles are congruent.

Explanation of Solution

Given information:

Theorem 4-2,

Two angles are congruent, then the sides opposite to those angles are congruent.

Calculation:

Equiangular: When all the interior angles are equal in measure or say all angles are congruent, then the figure is said to be equiangular.

The Isosceles Triangle Theorem: If Two angles are congruent, then the sides opposite to those angles are congruent.

Corollary 1: An equilateral triangle is also equiangular.

Given: $\Delta ABC$ is equiangular

Prove: $\Delta ABC$ is equilateral

Proof:

Statements	Reasons
1. $\Delta ABC$ is equiangular	Given
2. $\angle A\cong \angle C$	Definition of equiangular triangle
3. $\overline{AB}\cong \overline{BC}$	Converse of isosceles theorem
4. $\angle B\cong \angle C$	Definition of equiangular triangle
5. $\overline{AB}\cong \overline{AC}$	Converse of isosceles theorem
6. $\angle A\cong \angle B$	Definition of equiangular triangle
7. $\overline{BC}\cong \overline{AC}$	Converse of isosceles theorem
8. $\overline{AB}\cong \overline{BC}\cong \overline{AC}$	Transitive property
9. $\Delta ABC$ is equilateral	Definition of equilateral

Thus, the proof given above shows that how Corollary follows from the converse of isosceles triangle theorem.