# Questions on lines & angles – set 01

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.

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