Chapter 4.6, Problem 35PPS

To determine

To Prove:

Corollary $4.3$ .

Explanation of Solution

Proof:

Equilateral triangle corollary states that a triangle is equilateral if and only if it is equiangular” first show that if $△ABC$ is equiangular , then $△ABC$ is equilateral.

It is given that,

  $△ABC$ is equilateral.

  

Statements	Reasons
$△ABC$ is equiangular	Given
$\angle A\cong \angle B\cong \angle C$	Definition of equiangular triangle.
$\overline{BC}\cong \overline{AC}\cong \overline{AB}$	Converse of isosceles triangle theorem.
$△ABC$ is equilateral	Definition of equilateral triangle

Second show that if $△ABC$ is equilateral, then $△ABC$ is equiangular.

It is given that,

  $△ABC$ is equilateral.

  

Statements	Reasons
$△ABC$ is equiangular	Given
$\overline{BC}\cong \overline{AC}\cong \overline{AB}$	Definition of equiangular triangle.
$\angle A\cong \angle B\cong \angle C$	Isosceles triangle theorem.
$△ABC$ is equilateral	Definition of equilateral triangle