3.2.8.
An equilateral triangle has three congruent sides. Prove that an equilateral triangle is equiangular.
In this equilateral triangle, all three sides
are congruent: AB
≅
BC
≅
CA
.
The
Isosceles Triangle Theorem has been
proved in class (points D, E, F, G, H, and I
are necessary only to prove the Isosceles
Triangle Theorem, not the Equilateral).
If
AB
=
AC
, then, from the
Isosceles Triangle Theorem, we
know that:
∠
ABC
≅
∠
ACB
.
If
BA
=
BC
, then,
from the Isosceles Triangle
Theorem, we know that
∠
BCA
≅
∠
BAC
.
If
CA
=
CB
, then from the Isosceles Triangle Theorem, we know that
∠
CBA
≅
∠
CAB
.
Therefore
∠
ABC
≅
∠
BCA
≅
∠
CAB
3.2.9.
Complete the proof of Theorem 3.2.8 by proving the converse of the proven statement.
The converse of the proven statement: An equilateral triangle has three congruent angles. Prove that an
equiangular triangle is equilateral.
.

In this equiangular triangle, all three angles
are congruent:
∠
ABC
≅
∠
BCA
≅
∠
CAB
. The
Isosceles Triangle Theorem has been proved
in class (points D, E, F, G, H, and I are