Question

in the given figure ABCD is a quadrilateral in which ab parallel DC and P is the midpoint of BC on producing AP and DC meet at Q prove that (i)AB=CQ,(ii) DQ=DC+AB

Answer 1

given that:
AB ll DC
so, AB ll DQ
so, angle BAQ = angle DQA (alternate angles)
or angle BAP = angle CQP ------(1)

Now, in triangle ABP and triangle QCP,
angle BAP = angle CQP (from (1))
angle BPA = angle CPQ (vertically opposite angles)
BP = CP (since P is the midpoint of BC)

so, triangle ABP congruent triangle QCP (by AAS congruency)
or AB = CQ (by CPCT)   [proved]  -----(2)

again, DQ = DC + CQ = DC + AB  (from (2)) [proved]


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Answer 2

Answer:

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