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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

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#### Solution

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)

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