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.