Question

In a trapezium ABCD, the two non-parallel sides are given as AD and BC. If
AD = BC and DAB = CBA. Then prove that AC = BD and BAC = DBA.​

Answer 1

Answer:

△ABD and △BAC,

AD=BC (Given)

∠DAB=∠CBA (Given)

AB=BA (Common)

∴△ABD≅△BAC (By SAS congruence rule)

∴BD=AC (By CPCT)

And, ∠ABD=∠BAC

Step-by-step explanation:

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Answer 2

Answer:

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.

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                                       

(iv)   Since, ΔABD ≅ ΔBAC

Then , ∠ABD = ∠BAC