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

All questions are to the standard of grade 09 math.

**Question 01**

Consider the below figure in which ∠BOD = 40 degree and ∠ AOC + ∠ BOE = 70 degree. Find measure of ∠BOE & ∠COE.

**Solution**

It’s given that ∠BOD = 40 degree.

Note that **∠BOD and ∠COA are vertically opposite angle**.

∠ BOD = ∠COA = 40 degree

It’s given in question that ∠ AOC + ∠ BOE = 70 degree.

Putting the value of ∠AOC we get;

40 + ∠ BOE = 70 degree

∠ BOE = 70 – 40

**∠ BOE = 30 degree**

We know that line AB is a straight line and adjacent angle in a straight line measures 180 degree.

∠ AOC + ∠ COE + ∠ BOE = 180

40 + ∠COE + 30 = 180

∠COE = 180 – 70

∠COE = 110 degree

Hence, **measure of ∠COE is 110 degree.**

**Question 02**

In the below figure ∠PQR = ∠PRQ. Prove that ∠PQS = ∠PRT.

**Solution**

We know that in straight line adjacent angle measures 180 degree.

∠ SQP + ∠PQR = 180 degree

Similarly QRT is also a straight line. So we can write;

∠QRP + ∠TRP = 180 degree

Combining the above two expression.

∠ SQP + ∠PQR = ∠QRP + ∠TRP

We know that;

∠PQR = ∠PRQ ( given )

Rewriting the above expression;

∠ SQP + ∠PQR = ∠PQR + ∠TRP

Cancelling ∠PQR on both sides.

∠ SQP = ∠TRP

Hence Proved.

**Question 03**

Consider the below figure in which line PQ is a straight line. The line OR intersect PQ at right angle. Prove that \mathtt{\angle ROS=\frac{1}{2}( \angle QOS-\angle POS)}

**Solution**

Taking the right side of the expression;

= 1/2 (∠QOS – ∠POS)

∠QOS can be written as follows;

∠QOS = ∠QOR + ∠ROS

Similarly ∠POS is written as follows;

∠POS = ∠POR – ∠SOR

Putting both the values in main expression.

= 1/2 ( ∠QOR + ∠ROS – ( ∠POR – ∠SOR) )

= 1/2 ( ∠QOR + ∠ROS – ∠POR + ∠SOR)

= 1/2 ( 90 + ∠ROS – 90 + ∠SOR)

= 1/2 ( 2∠ROS )

= ∠ROS

Hence proved the given expression.

**Question 04**

Consider the below figure in which ∠XYZ = 64 degree. Note that ray YQ bisect ∠ZYP. Find the measure of ∠XYQ and ∠QYP.

**Solution**

Note that XYP is a straight line.

We know that in straight line adjacent angle measures 180 degree.

∠XYZ + ∠ZYP = 180 degree

64 + ∠ZYP = 180

∠ZYP = 180 – 64

∠ZYP = 116 degree

We know that YQ bisect ∠PYZ.

∠PYQ = ∠QYZ

∠ZYP can be written as;

∠PYQ + ∠QYZ = 116

2 ∠PYQ = 116

∠PYQ = 58 degree

Now we have got all the required data, let’s calculate the value of ∠XYQ.

∠XYQ + ∠QYP = 180

∠XYQ + 58 = 180

∠XYQ = 180 – 58

∠XYQ = 122 degree

Hence, we got all the required values.

**Question 05**

In the below figure, line PQ & RS intersect each other at point O. If ∠POR : ∠ROQ = 5 : 7, find the measure of all the angles.

**Solution**

Note that PQ is a straight line and we know that adjacent angle in straight line measures 180 degree.

∠POR + ∠ROQ = 180 degree

It’s given that, ∠POR : ∠ROQ = 5 : 7

Let ∠POR = 5x and ∠ROQ = 7x

Putting this value in above equation;

5x + 7x = 180 degree

12x = 180

x = 180 / 12

x = 15

Putting the value of x to find angle measurement.

∠POR = 5x

∠POR = 12 (5) = 60 degree

Similarly ∠ ROQ = 7x

∠ ROQ = 7(12) = 72 degree.

∠POS = ∠ ROQ = 72 (vertically opposite angle)

∠ POR = ∠ SOQ = 60 degree

Hence, we got the value of all angles.