In the figure (not to scale) AB = CD, angle DAP = angle DPA and A,B,C,D are concycli. If angle ADC = 80°, then find angle ACD.

Using Linear Pair:


Because Angle DPA = DAP , and we know that ADP is 100,
i am considering DPA as value “x”
so 2x + 100 = 180 (angle sum property of triangle)
x = 40*

Since we know that ABP is a straight line , we can derive that angle “A”’s total value is 90*
let us keep A = x + y (new variable)
                    90 = 40 + y
                    Y= 50*

Using Angle sum property,
A + C + D = 180
Since Triangle ACD is Linear , A = C
So, 50+80+C = 180
130 + C = 180

So finally , as a result , we get angle C = 50*.

PROVING C = 50* : 
As Triangle ACD is linear , A and C are of same value
I am considering x as value of both A and C.

So 2x+80=180
     2x = 100
     x = 50

Hope this helps you! Glad to help ;)