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

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In ΔABD and ΔBAC, AD = BC ∠DAB = ∠CBA AB = AB (common side) ∴ ΔABD ≅ ΔBAC by SAS Test of congruency ∴ BD = AC (corressponding sides of congruent traingles) ∴ ∠ABD = ∠BAC (corressponding angles of congruent traingles)
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In ΔABD and ΔBAC,

AD = BC

∠DAB = ∠CBA

AB = AB (common side)

∴ ΔABD ≅ ΔBAC by SAS Test of congruency

∴ BD = AC (corressponding sides of congruent traingles)

∴ ∠ABD = ∠BAC (corressponding angles of congruent traingles)

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