Chapter 4.4, Problem 10CE

To determine

To explain: the corollary an equilateral triangle has three $60°$ angles follows from corollary an equilateral triangle is also equiangular.

Explanation of Solution

Calculation:

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.

Corollary 2: An equilateral triangle has three $60°$ angles

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

Prove: y $m\angle A=m\angle B=m\angle C=60°$

Proof:

Statements	Reasons
1. $\overline{AB}\cong \overline{AC}$	Given
2. $\angle A\cong \angle B\cong \angle C$	Corollary 1
3. $m\angle A=m\angle B=m\angle C$	Definition of congruent angles
4. $m\angle A=m\angle B=m\angle C=180°$	Sum of angles of a triangle is   5. $180°$
6. $\begin{array}{c}m\angle A+m\angle A+m\angle A=180°\\ 3\times m\angle A=180°\\ m\angle A=\frac{180°}{3}=60°\end{array}$	Substitution property
7. Similarly   $m\angle B=m\angle C=60°$	Apply in same manner as in step 10

Thus, the proof given above shows that how Corollary 2 follows from Corollary 1.