Sides Opposite Equal Angles are Equal


 
 
Concept Explanation
 

Sides Opposite Equal Angles are Equal

Theorem: Angles opposite to two equal sides of a triangle are equal.

Given: A large Delta ABC in which  large angle B=angle C

To Prove:  AB = AC

Construction: Draw the bisector of large angle A  and Let it meet BC at D.

Proof:  In large large Delta sABD;and ;ACD, we have

               large angle B=angle C                              [Given]

          large angle BAC=angle CAD               [By construction] 

            AD = AD                                             [ Common ]

So, by AAS criterion of congruence, we have

         large Delta ABDcong Delta ACD

large Rightarrow ;;;AB=AC                         [C.P.C.T]

Illustration: Prove that measure of each angle of an equilateral triangle is 60^{circ}.

Solution:    Let Delta ABC be an equilateral triangle. Then, AB = BC = CA

Since angles opposite to equal sides of a triangle are equal.

therefore              AB = BC and BC = CA

Rightarrow ;;;angle C=angle A and           Rightarrow ;;;angle A=angle B

Rightarrow ;;;angle A=angle B=angle C

But,     Rightarrow ;;;angle A+angle B+angle C=180^{circ}     [ Angle Sum Property ]

therefore      angle A+angle A+angle A=180^{circ}

Rightarrow ;;;3angle A=180^{circ};;Rightarrow ;;angle A=60^{circ}

Hence,   angle A=angle B=angle C=60^{circ}

Sample Questions
(More Questions for each concept available in Login)
Question : 1

In the above figure, if angle DAC=angle CAB and angle DCA=angle ACB, then which of the following statement is true?

Right Option : C
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Explanation
Question : 2

In the figure given below, if angle BAC=angle EDF, then which statement is true?

Right Option : B
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Explanation
Question : 3

In the following figure, AB = AC, D is the point in the interior of bigtriangleup ABC such that angle DBC=angle DCB. Then which of the following statement is true?

Right Option : A
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Explanation
 
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