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Explain 2
Proving the Equilateral Triangle Theorem
and Its Converse
An equilateral triangle is a triangle with three congruent sides.
An equiangular triangle is a triangle with three congruent angles.
Equilateral Triangle Theorem
If a triangle is equilateral, then it is equiangular.
Example 2 Prove the Equilateral Triangle Theorem and its converse.
Step 1 Complete the proof of the Equilateral Triangle Theorem.
Given:
_
AB ≅
_
AC ≅
_
BC
Prove: ∠
A ≅ ∠B ≅ ∠C
Given that
_
AB ≅
_
AC we know that ∠B ≅ ∠ by the
.
It is also known that ∠A ≅ ∠B by the Isosceles Triangle Theorem, since .
Therefore, ∠A ≅ ∠C by .
Finally, ∠A ≅ ∠B ≅ ∠C by the Property of Congruence.
The converse of the Equilateral Triangle Theorem is also true.
Converse of the Equilateral Triangle Theorem
If a triangle is equiangular, then it is equilateral.
Step 2 Complete the proof of the Converse of the Equilateral Triangle Theorem.
Given: ∠A ≅ ∠B ≅ ∠C
Prove:
_
AB ≅
_
AC ≅
_
BC
Because ∠B ≅ ∠C,
_
AB ≅ by the
.
_
AC ≅
_
BC by the Converse of the Isosceles Triangle Theorem because
≅ ∠B.
Thus, by the Transitive Property of Congruence, , and therefore,
_
AB ≅
_
AC ≅
_
BC .
Reflect
4. To prove the Equilateral Triangle Theorem, you applied the theorems of isosceles
triangles. What can be concluded about the relationship between equilateral triangles
and isosceles triangles?
A
B C
A
B C
Module 7
330
Lesson 2
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