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 in the figure).Show that (i)△DAP≅△EBP and (ii)AD=BE
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Solution
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Given:P is the mid-point of AB,
So,AP=BP ......(1)
∠BAD=∠ABE ......(2)
∠EPA=∠DPB ......(3)
To prove:(i)△DAP≅△EBP
(ii)AD=BE
Proof:Since ∠EPA=∠DPB
We add ∠DPE both sides, we get
∠EPA+∠DPE=∠DPB+∠DPE
∠DPA=∠EPB ......(4)
In △DAP and △EBP,
∠DAP=∠EBP from (2)
AP=BP from (1)
∠DPA=∠EPB from (4)
∴△DAP≅△EBP by ASA congruence rule
∴AD=BE by C.P.C.T
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