Angle opposite to longer side is greater


 
 
Concept Explanation
 

Angle Opposite to Longer Side is Greater

Theorem: In a triangle the greater angle has the longer side opposite to it.

Given:  A Delta ABC in which angle ABC;>angle ACB

To Prove: AC>AB.

Proof:   In Delta ABC, we have the following three possibilities.

(i)AC=AB          (ii)AC<AB         (iii)AC>AB.

Out of these three possibilities exactly one must be true.

Case I:     When AC = AB

        As          AC = AB      Rightarrow ;;;angle ABC=angle ACB        [Angles opp. to equal sides are equal]

 But as   angle ABC;>angle ACB is given it contradicts our assumption.

therefore            ACneq AB

CASE II:  When   AC <AB

Rightarrow ;;;angle ACB>ABC         [because  Longer side has the greatest angle opposite to it]

This also contradicts the given statement.

Thus, we are left with the only possibility,  AC>AB, which must be true.

Hence,    AC>AB.

Sample Questions
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Question : 1

In the given figure, the measure of _____________ is greatest.

Right Option : B
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Explanation
Question : 2

In the following figure, PQRS is a quadrilateral. PQ is the longest side and RS is its shortest side. Then the relation between angle R and angle P is

Right Option : C
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Explanation
Question : 3

In the following figure, AB and CD are respectively the smallest and longest sides of a quadrilateral ABCD. Then   the relation between the angle B;and;angle D  is

Right Option : A
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Explanation
 
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