2. ABCD is a quadrilateral in which AD = BC and ∠DAB = ∠CBA (see Fig. 7.17). Prove that (i) ΔABD ≅ ΔBAC (ii) BD = AC (iii) ∠ABD = ∠BAC.
Solution: The given parameters from the questions are ∠DAB = ∠CBA and AD = BC. (i) ΔABD and ΔBAC are similar by SAS congruency as AB = BA (It is the common arm) ∠DAB = ∠CBA and AD = BC (These are given in the question) So, triangles ABD and BAC are similar i.e. ΔABD ≅ ΔBAC. (Hence proved). (ii) It is now known that ΔABD ≅ ΔBAC so, BD = AC (by the rule of CPCT). (iii) Since ΔABD ≅ ΔBAC so, Angles ∠ABD = ∠BAC (by the rule of CPCT).
