# Questions on parallel lines & transversal

Given below are collection of questions related to lines and angles.

All questions are to the standard of grade 09.

Question 01
In the below image line AB is parallel to CD. Also ∠FED = 90 degree and ∠GED = 126 degree. Find the measure of angle ∠AGE, ∠GEF and ∠FGE.

Solution
Since parallel lines AB & CD are intersected by transversal GE, the alternate interior angles are equal.

∠AGE = ∠GED

∠AGE = 126 degree.

∠GED can be written as;
∠GED = ∠GEF + ∠FED

126 = ∠GEF + 90

∠GEF = 126 – 90

∠GEF = 36 degree

When parallel lines are intersected by transversal then sum of same side interior angles measures 180 degree.

∠FGE + ∠GED = 180

∠FGE + 126 = 180

∠FGE = 180 – 126

∠FGE = 54 degree

Hence, value of all angles are calculated.

Example 02
In the below image PQ is parallel to line ST. Find the measure of angle ∠QRS.

Solution
Extend the line QM as shown in below image.

Here ST & PM are parallel lines with SR are transversal.

Corresponding angles are equal.
∠ 2 = 130 degree

QM is a straight line and adjacent angle in straight line measure 180 degree.
∠3 + ∠2 = 180

∠3 = 180 – 130

∠ 3 = 50 degree

Similarly, ∠4 + 110 = 180

∠4 = 180 – 110

∠ 4 = 70 degree

QOR is a triangle and sum of all angle of triangle measure 180 degree.
∠4 + ∠3 + ∠QRO = 180

70 + 50 + ∠QRO = 180

∠QRO = 180 – 120

∠QRO = 60 degree

Hence, the required angle measures 60 degree.

Question 03
In the below figure line AB & CD are parallel to each other. Find the value of angle x and y.

Solution
We know that when parallel line is intersected by transversal then alternate interior angles are equal.

Here AB & CD are parallel line intersected by transversal PQ.

∠x = ∠APQ

∠x = 50 degree

Similarly parallel lines AB & CD are intersected by transversal PR.
Again alternate interior angles are equal.

∠APR = ∠ PRD

50 + ∠y = 127

∠y = 127 – 50

∠y = 77 degree.

Hence, we got both the values.

Question 04
In the below figure lines P & Q and lines R & S are parallel to each other respectively. Given below are values of some angles.
∠1 = 3x + 15
∠2 = 4x – 5
∠3 = 5y

Find the values of x and y

Solution
Consider parallel lines P & Q intersected by transversal R.

Here ∠1 and ∠2 are corresponding angles which are equal.

∠1 = ∠ 2

3x + 15 = 4x – 5

4x – 3x = 15 + 5

x = 20

Consider parallel lines R & S intersected by transversal Q.
Here ∠2 and ∠3 are equal corresponding angles.

∠2 = ∠ 3

4x – 5 = 5y

4(20) – 5 = 5y

80 – 5 = 5y

75 = 5y

y = 15

Hence, we got the values of x and y.

Question 05
In the below figure, find the value of x and y.
Note that line AB is parallel to CD.

Solution
Consider line AB & CD intersected by transversal AD.
When parallel lines are intersected by transversal, the sum of interior angle on same side measures 180 degree.

∠A = ∠ D

90 = 3y + 18

3y = 90 -18

3y = 72

y = 24

Similarly consider parallel lines AB & CD intersected by transversal BC.
Here ∠B and ∠C are interior angle on same side.

∠B + ∠C = 180

15x + 30 + 10x = 180

25x = 180 – 30

25x = 150

x = 6

Hence, we got the value of x and y.

Question 06
In the below figure line P & Q are parallel to each other. Find the value of x, y and z.

Solution
Line Q is a straight line and sum of all adjacent angle in straight line measures 180 degree.

2x + 90 + x = 180

3x + 90 = 180

3x = 180 – 90

3x = 90

x = 30

Now consider parallel lines PQ intersected by transversal OS.
In this case, we know that sum of interior angle on same side measures 180 degree.

(2x + 90) + 2y = 180

2(30) + 90 + 2y = 180

60 + 90 + 2y = 180

150 + 2y = 180

2y = 180 – 150

2y = 30

y = 15

Line P is a straight line and sum of alternate angle on straight line measure 180 degree.

2y + z = 180

2(15) + z = 180

30 + z = 180

z = 180 -30

z = 150

Hence, we calculated the values of all variables.

Question 07
In the below figure line AB & CD are parallel to each other.
Find the value of variable x and y.

Solution
Line AB & CD are parallel intersected by transversal AD.
In this case sum of alternate angle on same side measures 180 degree.

( 3x + 2y ) + 4y = 180

3x + 6y = 180

3 (x + 2y) = 180

x + 2y = 60

Similarly AB & CD are parallel lines intersected by transversal CA.
In this case, vertically opposite angles are equal.

3x = 5x – 20

20 = 5x – 3x

2x = 20

x= 10

Putting the value of x in equation x + 2y = 60

10 + 2y = 60

2y = 50

y = 25

Hence, we got the value of x and y.

Question 08
In the below figure, line P & Q are parallel to each other.
Find the value of variable x.

Solution
To solve the question, draw a line M parallel to P & Q and passing through ∠x.

Line P & M are parallel lines intersected by transversal RO.

∠RON = 62 degree (alternate interior angle)

Similarly line Q & M are parallel to each other intersected by transversal OS.
We know that in this case, sum of of interior angle on same side measures 180 degree.

∠NOS + 144 = 180

∠NOS = 180 -144

∠NOS = 36 degree

Now let’s calculate the ∠ROS

∠ROS = ∠RON + ∠NOS

∠ROS = 62 + 36

∠ROS = 98 degree

Hence, value of x = 98 degree