# Questions on lines & angles – set 02

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

All the questions are provided with detailed solution.

The questions are to the standard of grade 09.

Question 01
In the below image PQ & RS are two parallel lines. If ∠QXM = 135 degree and ∠RYM = 40 degree then find the measure of ∠XMY.

Solution
To solve the question, draw an imaginary line AB parallel to PQ & RS.

PQ & AB are parallel line with XM as transversal.
We know that for parallel lines, interior angles on same side of transversal sum to 180 degree.

∠QXM + ∠XMB = 180

135 + ∠XMB = 180

∠XMB = 180 – 135

∠XMB = 45 degree

Similarly AB & RS are parallel lines with MY as transversal.
We know that when parallel line is intersected by transversal then alternate interior angles are equal.

∠ RYM = ∠ BMY = 40 degree

Combining both the above angles.
∠XMB + ∠ BMY = ∠XMY

40 + 45 = ∠XMY

∠XMY = 85 degree

Hence, 85 degree is the answer.

Question 02
In the below figure, transversal AD intersect line PQ & RS. Here line BE and CG are parallel to each other & bisect ∠ABQ and ∠BCS respectively. Prove that PQ & RS are parallel lines.

Solution
It’s given that BE & CG are parallel lines and AD is a transversal.
We know that when parallel lines is intersected by transversal then corresponding angles are equal.

∠ABE = ∠ BCG

Since BE & CG are angle bisector of ∠ABQ & ∠BCS respectively. We can write;
∠ABE = 1/2 (∠ABQ) and ∠BCG = 1/2 (∠BCS)

Putting both the values in above expression;
∠ ABE = ∠ BCG

1/2 (∠ABQ) = 1/2 (∠BCS)

Cancelling the common number from both sides, we get;
∠ABQ = ∠BCS

These are corresponding angle between lines PQ & RS intersected by AD as transversal.
If corresponding angles are equal and the given lines are parallel to each other.

Hence both lines PQ & RS are parallel.

Question 03
In the below figure lien AB, CD & EF are parallel to each other. Also ∠DEF = 55 degree and ∠BAE = 90 degree. Find the measure of angle x, y and z.

Solution
Here AB & EF are parallel lines and AE is a transversal. In this case angles on same side of transversal adds to 180 degree.

∠BAC + ∠FEC = 180

90 + ∠FEC = 180

∠ FEC = 90 degree

From the above figure we can say that ∠FEC is made of ∠z and ∠FED.
∠FEC = ∠z + ∠FED

90 = ∠z + 55

∠z = 90 – 55

∠z = 35 degree.

We know that CD||EF and ED is a transversal.
In his case angle on same side of transversal measure 180 degree.

∠y + 55 = 180 degree

∠y = 180 – 55

∠y = 125 degree

Note that ∠x & ∠y are corresponding angles.

∠x = ∠y

∠x = 125 degree

Hence, all the required angles are calculated.

Question 04
In the below figure, line AB & CD are parallel to each other. Find the value of ∠x and ∠y.

Solution
Since AB & CD are parallel line intersected by transversal, vertically opposite angles are equal.

∠y = 130 degree

Also in this case, alternate interior angles are also equal.
∠x = ∠y

∠x = 130 degree

Question 05
In the below figure line AB, CD & EF are parallel lines. If ratio of angle y : z is 3 : 7, then find the measure of angle y.

Solution
Its given that angle y : z is 3 : 7

Let ∠y = 3a and ∠z = 7a

We know that when parallel lines are intersected by transversal then vertically opposite angles are equal.

∠ y = ∠DOM = 3a

Also sum of interior angle on same side measure 180 degree.

∠DOM + ∠z = 180

3a + 7a = 180

10a = 180

a = 18 degree.

Now let’s calculate value of ∠y.
∠ y = 3a

∠ y = 3(18) = 54 degree.

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