We can begin by reviewing the Isosceles Triangle Theorem and its corollary.
Isosceles Triangle Theorem
First Corollary
If two sides of a triangle are congruent, then the angles opposite those sides are congruent.
If a triangle is equilateral, then the triangle is equiangular.
To show how the corollary follows form the Isosceles Triangle Theorem, let's prove the it using the theorem.
Given: XY ≅ YZ ≅ ZX
Prove: ∠ X ≅ ∠ Y ≅ ∠ Z
Notice that we know that XY ≅ YZ. Thus, by the Isosceles Triangle Theorem, the angles opposite those sides are congruent.
∠ X ≅ ∠ Z
We also know that YZ ≅ ZX, so by the same theorem we can conclude that ∠ X ≅ ∠ Y. Combining this, we get the following statement.
∠ Y ≅ ∠ X ≅ ∠ Z
This is what we wanted to show!
Second corollary
We can begin by reviewing the Converse of the Isosceles Triangle Theorem and its corollary.
Converse of the Isosceles Triangle Theorem
Second Corollary
If two angles of a triangle are congruent, then the sides opposite those angles are congruent.
If a triangle is equiangular, then the triangle is equilateral.
To show how the corollary follows form the Converse of the Isosceles Triangle Theorem, let's prove the it using the theorem.
Given: ∠ X ≅ ∠ Y ≅ ∠ Z
Prove: XY ≅ YZ ≅ ZX
Notice that we know that ∠ X ≅ ∠ Z. Thus, by the Converse of the Isosceles Triangle Theorem, the sides opposite those angles are congruent.
XY ≅ YZ
We also know that ∠ Z ≅ ∠ Y, so by the same theorem we can conclude that XY ≅ ZX. Combining this, we get the following statement.
ZX ≅ XY ≅ YZ
This is what we wanted to show!
