Introduction to Angle-side-angle congruence postulate: 

Two triangles are congruent if two angles and the included side of  one triangle are equal to the corresponding two angles and the included side of the other triangle.

 

Let us prove the angle side angle congruence postulate.

Given: Two `Delta ABC and ` `Delta` DEF such that  m<B = m<E, m<C = m<F, and

 BC = EF.

To prove: `Delta ABC ~= Delta DEF `

proof of the triangle

Proof: There are three possibilites,

Case 1: When AB = DE

In this case, we have,

     AB = DE

    m<B = m<E     ( given)

     BC = EF,          (given)

So, by SAS congruence we have,

              `Delta ABC ~= Delta DEF`

Case 2: When AB < ED

 In this case, take a point G on ED such that EG = AB. Join GF.

 Now, in `Delta` ABC and `Delta` GEF, we have

     AB = GE               ( by supposition)

    m<B = m<E         (given)

     BC = EF               (given)

So, by SAS congruence, we have

     `Delta ABC ~= Delta GEF`

  `=>` <ACB = <GFE

but , <ACB = <DFE        (given)

  `:.` <GFE = <DFE

 This is possible only when ray FG coincides with ray FD or G coincides with D.

 Therefore, AB must be equal to DE

Thus, in `Delta ABC ` and `DeltaDEF` , we have,

   AB = DE         ( as proved)

   m<B = m<E    (given)

   BC = EF

So, by SAS congruence, we have

    `Delta ABC ~= Delta DEF`

Case 3:  When AB > ED

 In this case, we take a point G on ED produced such that EG = AB. Join GF.

Now, proceeding exactly on the same lines as in case 2, we can prove that,

     `Delta ABC ~= Delta DEF`

Some Solved Example Based on Angle side Angle Congruence Postulate

1) in fig , diagonal AC of the quadrilateral ABCD bisects the angles A and C. Prove that AB = AD and CB =CD

                           example 1

Solution :Since diagonal AC bisects the angles A and C

`rArr`  <BAC  = <DAC and <BCA = <DCA

In triangles ABC and ADC, we have

             <BAC = <DAC                    (given)

             <BAC = <DCA                    (given)

    and    AC =AC                             (Common)

So, by ASA (angle side angle congruence postulate) , we have

        `Delta` BAC    `~=`  `Delta`  DAC

         `rArr` BA  =DA   and CB = CD

 

 

Practice Problems Based on Angle side Angle Congruence Postulate :

1) In two right triangles, one side and an acute angle of one triangle are equal to one side and the corresponding acute angleof the otehr. Prove the two triangles are congruent.

2)Two lines AB nd CD intersect at O such that BC is equal and parallel to AD .Prove that the lines AB and CD bisect at O.