Question

AB is a line segment and P is its mid-point. D and E are points on the same side of AB such that ∠ BAD = ∠ ABE and ∠ EPA = ∠ DPB (see Fig). Show that 

(i) ∆ DAP ≅ ∆ EBP 

(ii) AD = BE

Answer 1

It is given that EPA = DPB 

∠EPA +∠ DPE = ∠DPB + ∠DPE 

∠DPA = ∠EPB

 In ∆DAP and ∆EBP,

 ∠DAP = ∠EBP (Given) 

AP = BP (P is mid-point of AB) 

∠DPA = ∠EPB (From above) 

∠∆DAP ∠∆EBP (ASA congruence rule) 

∴ AD = BE (By CPCT)