Chapter 4.4, Problem 9CE

To determine

To explain: the corollary an equilateral triangle is also equiangular.

Explanation of Solution

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 sides of a triangle are congruent, then the angles opposite to those sides are congruent.

Corollary 1: An equilateral triangle is also equiangular.

Given: $\Delta ABC$ is equilateral such that $\overline{AB}\cong \overline{BC}\cong \overline{CA}$

Prove: $\Delta ABC$ is equiangular

Proof:

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

Thus, the proof given above shows that how corollary 1 follows from the Isosceles triangle theorem.