ABCD is a quadrilateral in which AD = BC and ∠ DAB = ∠ CBA (see Fig. 7.17 ). Prove that
(i) ∆ ABD ≅ ∆ BAC
(ii) BD = AC
(iii) ∠ ABD = ∠ BAC.

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Detailed Solution

Given: AD = BC and ∠DAB = ∠CBA

To Prove: i) △ ABD ≅ △ BAC ii) BD = AC iii) ∠ABD = ∠BAC

ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that i) △ ABD ≅ △ BAC ii) BD = AC iii) ∠ABD = ∠BAC

(i) In △ABD and △BAC,

AD = BC (Given)

∠DAB = ∠CBA (Given)

AB = BA (Common)

∴ △ABD ≅ △BAC (By SAS congruence rule)

(ii) Since △ABD ≅ △BAC,

∴ BD = AC (By CPCT)

(iii) Since △ABD ≅ △BAC,

∠ABD = ∠BAC (By CPCT)

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