Question 1096896
<br>I have seen many students who like to solve this problem by the method shown by the other tutor.  I find it easier to see the solution by a different method, which I have also seen many students use.  Use the one which makes the most sense to you.<br>
Here is the diagram of the problem from the other tutor:<br>


{{{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)
)}}}<br>
And here is what I do with the diagram to solve the problem.<br>
Draw line EF through P parallel to AB and CD.
{{{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(4.2,3.65,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),
line(-1,3.15,12,3.15),
locate(1,3.65,E),
locate(11,3.65,F)
)}}}<br>
Now angle XPF is congruent to angle PXA, and angle YPF is congruent to angle PYC.  So the measure of angle P is just 35 degrees plus 42 degrees = 77 degrees.