# What is one proof of the converse of the Isosceles Triangle Theorem?

See explanation.

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One proof of the converse of the Isosceles Triangle Theorem can be done by assuming that in triangle ABC, side AB is equal to side AC (the triangle is isosceles), and then proving that the angles opposite these sides are equal (angle A = angle C).

Proof: Given: Triangle ABC is isosceles, AB = AC. To Prove: angle A = angle C.

Proof: Since triangle ABC is isosceles, AB = AC. Let's assume that angle A is not equal to angle C. Without loss of generality, let angle A be greater than angle C.

Construct angle ADC congruent to angle ABC and AD congruent to AB.

Now, in triangle ADC and triangle ABC: AD = AB (given) angle A = angle D (construction) angle ADC = angle ABC (construction)

By the Side-Angle-Side (SAS) congruence criterion, triangle ADC is congruent to triangle ABC.

However, this implies that AC = AD, which contradicts the fact that AD = AB and AB = AC.

Therefore, our assumption that angle A is not equal to angle C is incorrect, and hence, angle A must be equal to angle C. This proves the converse of the Isosceles Triangle Theorem.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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