Part (i)
We're told that which means angle DAB is 60 degrees. This angle is cut into two equal halves of 30 degrees each (for angles DAP and PAB) due to segment AP being a bisector of angle DAB.
Since ABCD is a parallelogram, the adjacent angles A and B are supplementary. A = 60 leads to...
A+B = 180
B = 180 - A
B = 180-60
B = 120
Angle ABC is 120 degrees which splits in half to get 60. The angles PBA and PBC are 60 degrees each.
Focus on triangle PAB. We found A = 30 and B = 60 earlier. That must mean:
P+A+B = 180
P+30+60 = 180
P+90 = 180
P = 180-90
P = 90
Triangle PAB is a right triangle with angle P being the 90 degree angle. This is another way of saying angle APB is 90 degrees.
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Part (ii)
From the previous part, we know that angles DAP and PAB are 30 degrees each. The alternate interior angles DPA and PAB are equal because we have a parallelogram, so angle DPA is also 30 degrees.
For triangle DPA, the base angles A and P are congruent (30 degrees each). This leads immediately to the fact triangle DPA is isosceles. The congruent sides are opposite the congruent base angles.
Therefore, AD = DP
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Also from part (i), we found that angles PBA and PBC were 60 degrees each.
Since we have a parallelogram, the alternate interior angles PBA and BPC are congruent, meaning that angle BPC is also 60.
For triangle PBC, we have the interior angles of:
P+B+C = 180
60+60+x = 180
120+x = 180
x = 180-120
x = 60
Each interior angle of triangle PBC is 60 degrees. Triangle PBC is equilateral. By definition, equilateral triangles have all sides the same length.
Therefore, PB = PC = BC.
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Part (iii)
ABCD is a parallelogram with congruent opposite sides.
Let AD = BC = x
From part (ii), we found that AD and DP were the same length. So DP = x as well.
Also from part (ii), we found that BC = PC, so PC = x
We can then say:
DC = DP + PC
DC = x + x
DC = 2x
DC = 2*AD