Question

ABCD is a quadrilateral in which AD=BC and ∠DAB = ∠CBA. Prove that a) ΔABD ≅ ΔBAC b) BD = AC c)∠ABD=∠BAC

Answer 1

Answer:

here is you're answer friend.

Step-by-step explanation:

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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Answer 2

Answer:

As per given in the question,

∠DAB = ∠CBA and AD = BC.

(i) ΔABD and ΔBAC are similar by SAS congruency as

AB = BA (common arm)

∠DAB = ∠CBA and AD = BC (given)

So, triangles ABD and BAC are similar

i.e. ΔABD ≅ ΔBAC. (Hence proved).

(ii) As it is already proved,

ΔABD ≅ ΔBAC

So,

BD = AC (by CPCT)

(iii) Since ΔABD ≅ ΔBAC

So, the angles,

∠ABD = ∠BAC (by CPCT).