Two lines and are intersected by a transversal at points and respectively

MATHEMATICS · Multi-part · CBSE · Mathematics · Lines and Angles · Lines and Angles

Problem

Two lines

k

and

l

are intersected by a transversal

d

at points

P

and

Q

respectively. The two angles marked at

P

and

Q

are situated as alternate interior angles. Their measures are given by

\angle APQ = (3x + 18)^{\circ}

and

\angle BQP = (5x - 42)^{\circ}

. (Note:

A

and

B

are points on lines

k

and

l

respectively, chosen so that

\angle APQ

and

\angle BQP

lie between the two lines.) (a) If

k \parallel l

, find the value of

x

. (b) Without assuming that the lines are parallel, determine whether

k

is parallel to

l

. Give a complete proof based on your result from part (a). (c) Once the relationship between

k

and

l

is established, find the measure of all the angles at points

P

and

Q

. Show each step clearly.