Side opposite to larger angle is greater


 
 
Concept Explanation
 

Side Opposite to Larger Angle is Greater

Theorem: If two sides of a triangle are unequal, the longer side has greater angle opposite to it.

Given:  A Delta ABC in which AC>AB.

To Prove: angle ABC>angle ACB

Construction:  Mark a point D on AC such that AB = AD. Join  BD.

Proof:  In Delta ABD,  we have

                    AB = AD                                                   [By construction]

large Rightarrow ;;;angle ADB=angle ABD                          [because Angles opp. to equal sides are equal]    ...(i)

Now, consider Delta BCD. We find that angle ADB, is the exterior angle of Delta sBCD and an exterior angle is always greater than interior opposite angle. Therefore,

         angle ADB;>;angle DCB

Rightarrow ;;angle ADB;>;angle ACB                                        [because ;angle ACB=angle DCB]   ....(ii)

From (i) and (ii), we have

Rightarrow ;;;    angle ABD;>;angle ACB                                .....(iii)

But,   angle ABD  is a part of angle ABC.

therefore       angle ABC;>;angle ABD                             .....(iv)

From (iii) and (iv), we get

           angle ABC;>;angle ACB

Hence Proved

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

In the following figure, AB > AC. The relation between AB and AD is ?

Right Option : B
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Explanation
Question : 2

In a triangle ABC, angle A=90^{0}angle B=30^{0} and angle C=60^{0}. Then which side will be of greater length?

Right Option : B
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Explanation
Question : 3

In the following figure , PQ > PR. QS and RS are the bisectors of angle Q;and;angle R respectively. Find the relation between SQ and SR.

Right Option : C
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Explanation
 
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