Question 1096896
{{{drawing(450,300,-2,13,-1.5,8.5,
line(-2,0,13,0),line(-2,7,13,7),
arrow(8,0,12,0),arrow(8,7,12,7),
circle(4.5,3.15,0.1),locate(4.2,3.5,P),
red(arc(10,7,2,2,145,180)),locate(8.1,7.1,red(35^o)),
red(arc(8,0,2,2,180,222)),locate(6.2,0.8,red(42^o)),
line(4.5,3.15,8,0),line(4.5,3.15,10,7),
locate(9.8,7.5,X),locate(7.9,0,Y),
circle(-1,0,0.1),circle(11,0,0.1),
circle(-0.3,7,0.1),circle(12.5,7,0.1),
locate(12.3,7.6,B),locate(-0.5,7.6,A),
locate(-1.2,-0.1,C),locate(10.8,-0.1,D)
)}}}
 
Let us extend segment PX until it meets line CD at point {{{green(Z)}}} :
 
{{{drawing(450,300,-2,13,-1.5,8.5,
line(-2,0,13,0),line(-2,7,13,7),
arrow(8,0,12,0),arrow(8,7,12,7),
circle(4.5,3.15,0.1),locate(4.2,3.5,P),
red(arc(10,7,2,2,145,180)),locate(8.1,7.1,red(35^o)),
red(arc(8,0,2,2,180,222)),locate(6.2,0.8,red(42^o)),
green(line(0,0,10,7)),locate(-0.1,0,green(Z)),
line(4.5,3.15,8,0),line(4.5,3.15,10,7),
locate(9.8,7.5,X),locate(7.9,0,Y),
circle(-1,0,0.1),circle(11,0,0.1),
circle(-0.3,7,0.1),circle(12.5,7,0.1),
locate(12.3,7.6,B),locate(-0.5,7.6,A),
locate(-1.2,-0.1,C),locate(10.8,-0.1,D)
)}}}
 
Angle {{{PYC}}} is the same angle as {{{PYZ}}} (just called by a different name),
and measures {{{42^o}}} .
Angle {{{AXP}}} is the same angle as {{{AXZ}}} (just called by a different name),
and measures {{{35^o}}} .
Angle {{{XZD}}} is congruent to angle as {{{AXZ}}} ,
because they are alternate interior angles.
So, angle {{{XZD=PZY}}} measures {{{35^o}}} too.
The angle at P, angle {{{XPY}}} is an exterior angle to traingle {{{PYZ}}} ,
and as all exterior angles to a triangle,
its measure is the sum of the measures of the non-adjacent interior angles of the triangle
(in this case angles {{{PZY}}} and {{{PYZ}}} ).
 
So, the measure of angle {{{XPY}}} is {{{35^o+42^o=highlight(77^o)}}} .