Question

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Answer 1

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The Isosceles Triangle Theorem states: If two sides of a triangle are congruent, then angles opposite those sides are congruent.

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Answer 2

Theorem 1:

Angles opposite to the equal sides of an isosceles triangle are also equal.

Proof: Consider an isosceles triangle ABC where AC = BC.

We need to prove that the angles opposite to the sides AC and BC are equal, that is, ∠CAB = ∠CBA.

Isosceles Triangle

We first draw a bisector of ∠ACB and name it as CD.

Now in ∆ACD and ∆BCD we have,

AC = BC (Given)

∠ACD = ∠BCD (Byconstruction)

CD = CD (Common to both)

Thus, ∆ACD ≅∆BCD (By SAS congruence criterion)

So, ∠CAB = ∠CBA (By CPCT)

Hence proved.

Theorem 2:

Sides opposite to the equal angles of a triangle are equal.

Proof: In a triangle ABC, base angles are equal and we need to prove that AC = BC or ∆ABC is an isosceles triangle.

Isosceles Triangle Theorem 2

Construct a bisector CD which meets the side AB at right angles.

Now in ∆ACD and ∆BCD we have,

∠ACD = ∠BCD (By construction)

CD = CD (Common side)

∠ADC = ∠BDC = 90° (By construction)

Thus, ∆ACD ≅ ∆BCD (By ASA congruence criterion)

So, AC = BC (By CPCT)

Or ∆ABC is isosceles.

Hence proved.

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