Question

Abcd is a parallelogram pbe midpoint of ab and bisect

Answer 1

ol: ABCD is a parallelogram, in which ∠A = 60° ⇒ ∠B = 120° [Adjacent angles of a parallelogram are supplementary] ∠C = 60° = ∠A [Opposite angles of a parallelogram are equal] ∠D = ∠B = 120° [Opposite angles of parallelogram are equal] AP bisects ∠A ⇒ ∠DAP = ∠PAB = 30° BP bisects ∠B ⇒ ∠CBP = ∠PBA = 60° In ΔPAB, ∠APB = 90° [Angle sum property] In ΔPBC, ∠BPC = 60° [Angle sum property] In ΔADP, ∠APD = 30° [Angle sum property] In ΔPBC, ∠BPC = ∠CBP = 60° [Linear angles are supplementary] ⇒ BC = PC [Sides opposite to equal angles of a triangle are equal] -------- (1) In ΔADP, ∠APD = ∠DAP = 30° ⇒ AD = DP [Sides opposite to equal angles of a triangle are equal] But AD = BC [Opposite sides of parallelogram are equal] So, BC = DP -------- (2) From (1) and (2), we get DP = PC ⇒ P is the midpoint of CD.

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