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Properties of a Triangle

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Construct a triangle in which two sides are equal, say each equal to 3.5 cm and the third side equal to 5 cm in following fig. 

A triangle in which two sides are equal is called an isosceles triangle. So, ∆ ABC of above Fig. is an isosceles triangle with AB = AC. 

theorem

Theorem : Angles opposite to equal sides of an isosceles triangle are equal.  
Proof : We are given an isosceles triangle ABC in which AB = AC. We need to prove that ∠ B = ∠ C.  Let us draw the bisector of ∠ A and let D be the point of intersection of this bisector of 
∠ A and BC in following fig. 

In ∆ BAD and ∆ CAD,
AB = AC (Given) 
∠ BAD = ∠ CAD (By construction)
AD = AD (Common) 
So,  ∆ BAD ≅ ∆ CAD (By SAS rule) 
So,  ∠ ABD = ∠ ACD, since they are corresponding angles of congruent triangles. 
So,  ∠ B = ∠ C

Theorem : The sides opposite to equal angles of a triangle are equal.
This is converse of above theorem.

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