Question

n fig if\[AB\parallel CD\], \[\angle APQ = {40^ \circ }\]and \[\angle PRD = {118^ \circ }\]find x and y.

Answer 1

Hint: When two parallel lines are cut by a transversal, then the alternate interior angles are congruent, and their alternate exterior angles are also congruent.
Parallel lines are the line in a plane that is always at a distance apart, and they never intersect. Curves that do not touch each other and are at a minimum fixed distance, then they are said to be parallel.

Complete step-by-step answer:
In this question, line AB is parallel to line AD, and line PR is transversal, so by using the property of the parallel lines, we can say \[\angle APQ = \angle PQR\]since they are alternate angles and after getting the value of ‘x’ find the value of y by using the exterior angle of a triangle theorem.
\[AB\parallel CD\]
\[\angle APQ = {40^ \circ }\]
\[\angle PRD = {118^ \circ }\]
Here line PR is a transversal line; hence by using the congruent angle theorem of parallel lines, we can say that \[\angle APQ = \angle PQR\]
\[\therefore \angle PQR = x = {40^ \circ } - - - - (i)\]
Now in the\[\vartriangle PQR\], to find the value of y, use the exterior angle of a triangle theorem, which states the exterior angle of a triangle is equal to the sum of the two opposite interior angles,
Hence we can say
\[\angle PQR + \angle QPR = \angle PRD\]
Can also be written as
\[x + y = \angle PRD - - - - (ii)\]
Now substitute the values from equation (i) and (ii), we get:
\[
  40 + y = 118 \\
  y = 118 - 40 \\
   = {78^ \circ } \\
 \]
Hence the value of \[y = {78^ \circ }\]

Note: Alternate method to find the value of y is by using the total internal angle of a triangle theorem, which is equal to 180. Where \[\angle PRQ\]and \[\angle PRD\]are the angle of a straight line. Hence the sum of angles\[\angle PRQ\] \[\angle PQR\]and \[\angle QPR\]will be equal to 180.