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 the given figure). Show that 

(i) ΔDAP ≅ ΔEBP

(ii) AD = BE

Answer 1

It is given that ∠EPA = ∠DPB

∠EPA + ∠DPE = ∠DPB + ∠DPE

∠DPA = ∠EPB

In ΔDAP and ΔEBP,

∠DAP = ∠EBP (Given)

AP = BP (P is mid-point of AB)

∠DPA = ∠EPB (From above)

∴ ΔDAP ≅ ΔEBP (ASA congruence rule)

AD = BE (By CPCT)