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