Congruence of Triangles By AAS Criteria


 
 
Concept Explanation
 

Congruence of Triangles by AAS Criteria

Theorem : If any two angles and a non-included side of one triangle are equal to the corresponding angles and side of another triangle, then the two triangles are congruent.

Given: Two large Delta sABC  and DEF such that

              large angle A=angle D,angle B=angle E, BC=EF

To Prove:  large Delta ABCcong Delta DEF

Proof:  We have,

              large angle A=angle D;and;angle B=angle E

large Rightarrow ;;angle A+angle B=angle D+angle E

large because angle A+angle B+angle C=180^{circ};therefore angle A+angle B=180^{circ}-angle C

Similarly, large because angle D+angle E+angle F=180^{circ};therefore angle D+angle E=180^{circ}-angle F

large Rightarrow ;;180^{circ}-angle C=180^{circ}-angle F                              

Arrange the terms, we have

angle F-angle C=180^{0}-180^{0}

angle F-angle C=0

large Rightarrow ;;;angle C=angle F       .....................(i)

Thus, in large Delta sABC and DEF , we have

      large angle A=angle D,angle B=angle E  and large angle C=angle F

Now, in large Delta ABC  and  large Delta DEF, we have

              large angle B=angle E                                        [Given]

               BC = EF                                                 [Given]

and,        large angle C=angle F                                       [From (i)]

So, by ASA criterion of congruence, large Delta ABCcong Delta DEF

Hence Proved

 

Theorem: If two angles of a triangle are equal. then sides opposite to them are also 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 Delta sABD and ACD, we have

               large angle B=angle C               [Given]

          large angle BAC=angle CAD    [Each 90 degrees]                  AD = AD

So, by AAS criterion of congruence, we have

         large Delta ABDcong Delta ACD

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

Illustration: If bigtriangleup ABC is  an isosceles triangle with AB = AC. Prove that the perpendiculars from the vertices B and C to their opposite sides are equal.

Solution: In bigtriangleup ABC, we have

AB = AC                   [Given]

Rightarrow angle B=angle C                   ..........................(i)  [because Angles opposite to equal sides are equal]

Now, in bigtriangleup BCE;and;BCD, we have

angle B=angle C                                    [From (i)]

angle CEB=angle BDC                      [Each equal to 90^{0}]

and, BD = BC                                     [Common]

So, by AAS criterion of congruence, we have

bigtriangleup BCE=bigtriangleup BCD

Rightarrow BD=CE      [because Corresponding parts of congruent triangles are equal]

Hence, BD = CE

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

If the two triangles are congruent by AAS criteria. small angle ACB=angle DEF  and small angle CAB=angle EDF Then which sides are equal in both triangles.

Right Option : B
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Explanation
Question : 2

In figure given below, AD is a median and BL, CM are perpendiculars drawn from B and C respectively on AD and AD produced. Then bigtriangleup BDL;and ;bigtriangleup CDM are congruent  by which criteria?

Right Option : D
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Explanation
Question : 3

In the figure given below, AD is a median and BL,CM are perpendiculars drawn from B and C respectively on AD and AD produced. Then BL = CM by CPCT using which criteria on congruence?

Right Option : D
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Explanation
 
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