Question

"Question 2 ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (See the given figure). Prove that (i) ΔABD ≅ ΔBAC (ii) BD = AC (iii) ∠ABD = ∠BAC.

Class 9 - Math - Triangles Page 119"

Answer 1

Congruence of triangles:

Two ∆’s are congruent if sides and angles of a triangle are equal to the corresponding sides and angles of the other ∆.

 

In Congruent Triangles corresponding parts are always equal and we write it in short CPCT i e, corresponding parts of Congruent Triangles.

 

It is necessary to write a correspondence of vertices correctly for writing the congruence of triangles in symbolic form.

 

Criteria for congruence of triangles:

There are 4 criteria for congruence of triangles.

SAS( side angle side):

Two Triangles are congruent if two sides and the included angle of a triangle are equal to the two sides and included angle of the the other triangle.

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First use, SAS rule to show congruence of triangles and then use CPCT to show ii & iii parts.

Given:

In quadrilateral ABCD,

AD = BC &

∠DAB = ∠CBA

 

To Prove:

(i)              ΔABD ≅ ΔBAC

(ii)            BD=AC

(iii)           ∠ABD = ∠BAC

 

Proof:

i)

In ΔABD & ΔBAC,
AB = BA    (Common)
∠DAB = ∠CBA  (Given)
AD = BC (Given)

Hence, ΔABD ≅ ΔBAC.        

( by SAS congruence rule).

(ii) Since, ΔABD ≅ ΔBAC
Then, BD = AC                                       ( by CPCT)

(iv)   Since, ΔABD ≅ ΔBAC
Then , ∠ABD = ∠BAC                         (by CPCT)

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Hope this will help you.....

Answer 2

in triangle ABD and triangle BAC

AB=BC (GIVEN)
∠DAB = ∠ CBA (GIVEN)
AB = BA (COMMON)

THEREFORE, TRIANGLE ABD ≅  TRIANGLE BAC ( BY SAS)
THEREFORE, BD = AC (CPCT)

ALSO, ∠ ABD = BAC (CPCT)

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hope it is helpful
@ debeshee