Question

find x and y in the following figure

Answer 1

Construction: Make a triangle BCZ, where z is the mid point of RS. now the so formed shape is a square ABCZ.

All angles of a square are 90°, so <BAC=90°.
Also, all sides of square ABCZ are same, so AB=AC, <ABC=<BCA (Angles in an isosceles triangle).-(1)

so, we've <ABC+<BCA+<BAC=180°(ASP)
FROM 1,
2<BCA+90°=180°
2<BCA=90°
<BCA=45°=<ABC-(2)
___So, y=45°___

Also,<CBP=90°=<ABC+<ABP
FROM 2,
<CBP=45°

___So, x=45°___

Answer 2

PBCQ is a rectangle (pb // qc and pb = qc)
Hence, angleP = angleC =angle B = angle Q =90°
AngleQ + angleQCA + angleCAQ =180°(sum of angles of triangle)
90°+ 2 angle QCA = 180°( angles opposite to equal sides are also equal)
2 angle QCA = 90
Angle QCA = 45°

Angle QCA + y = 90°
45° +y = 90°
y = 45°

In triangle QCA and PAB
PB = QC
PA = AQ
Angle APB = angle QCA =90°
TRIANGLE QCA =~ TRIANGLE PBA
Angle PBA = angle QCA ( C.P.C.T)
x =45°