There are **two keywords to note** here:

(a) Same Side

(b) Interior

So, when two lines are intersected by a transversal, then the angle formed on **same side of the transversal** and in the **interior of the lines** are known as **Same Side Interior Angles**.

Observe the image below:

M & N are two lines intersected by transversal T

Here;

Angles 1 & 2 are Same Side Interior Angles

Angles 3 & 4 are also Same Side Interior Angles

**Same Side Interior Angle Theorem**

The theorem says that when the **two lines are parallel** and are intersected by the transversal then the set of **Same Side Interior Angles** are **Supplementary**.

Important Point to note:

(a) Line should be parallel

(b) Same Side Interior Angles are Supplementary

In the above figure;

M & N are two parallel lines intersected by transversal T.

In this case, according to Same Side Interior Angle Theorem;

⟹ \angle 1\ + \ \angle 2 = 180

⟹ \angle 3\ + \ \angle 4 = 180

**Proof of Same Side Interior Angle Theorem**

**Theorem****When two parallel lines are intersected by transversal, then the same side interior angles are supplementary**

**To Prove**

⟹ \angle 1\ + \ \angle 2 = 180

**Solution**

Since transversal T is a straight line, the sum of angle in the line will add to 180 degree

⟹ \angle 1\ + \ \angle 3 = 180 — eq(1)

Since both line M & N are parallel, we can say that:

\angle 2\ = \angle 3\ {Corresponding Angle} —–eq(2)

Putting eq(2) in eq(1), we get:

⟹ \angle 1\ + \ \angle 2 = 180

**Corollary of Same Side Interior Angle Theorem**

It says when the same side interior angles are supplementary, then the given lines are parallel to each other.

This concept is just converse of the theorem discussed above.**Example **

In the above figure, angle y and z are Same Side Interior Angle

\angle y\ = 104 degree

\angle z\ = 76 degree

Adding angle y and z, we get:

\angle y\ + \angle z\ = 180 degree

Since both the angles are supplementary.

Then using corollary of Same Side Interior Angle Theorem, the given lines are parallel to each other.

**Same Side Interior Angle Questions**

**(01) Find angle y, using Same Side Interior Angle Theorem**

Angle x and y are same side interior angles

Since the given lines are parallel;

⟹ \angle x\ + \angle y\ = 180 degree

⟹ 78 + \angle y\ = 180 degree

⟹ \angle y\ = 180 – 78

⟹ \angle y\ = 102 degree

**(02) Find the value of angle x in the below figure**

Angle x and Angle y are same side interior angles

Since both the lines are parallel;

⟹ \angle x\ + \angle y\ = 180 degree

⟹ \angle x\ = 180 – 67

⟹ \angle x\ = 113 degree

**(03) Find if the given lines M & N are parallel or not**

In the above figure, angle x and angle y are Same Side Interior Angles \angle x\ = 120 degree

\angle y\ = 56 degree

Adding both the angles we get:

\angle x\ + \angle y\ = 120 + 56

\angle x\ + \angle y\ = 176

Using Corollary of Same Side Interior Angles, we can say that:

Since both the angles are not supplementary, the lines M & N are not parallel to each other.

**(04) Find value of angle z using Same Side Interior Angle Theorem**

Since transversal T is a straight line.

The angle formed will add up to 180 degree

⟹ \angle x\ + \angle y\ = 180

⟹ \angle y\ =180 – 65

⟹ \angle y\ =115

From the image you can observe that angle y & z are same side interior angle

⟹ \angle y\ + \angle z\ = 180

⟹ \angle z\ = 180 – 115

⟹ \angle z\ = 65 degree

**Hence, 65 degree is the right answer**

**(05)** In the below image, line P & Q are parallel and line M & N are parallel to each other.

Find the value of **angle Z** using Same Side Interior Angle Theorem

Since line M & N are parallel to each other.

⟹ \angle x\ = \angle A\ { Corresponding Angles}

⟹ \angle A\ = 72 degree

Since N is a straight line

The sum of angles will be 180 degrees

⟹ \angle A\ + \angle B\ = 180

⟹ \angle B\ = 180 – 72

⟹ \angle B\ = 108 degree

Since line P & Q are parallel to each other, the same side interior angle will be supplementary

⟹ \angle B\ + \angle Z\ = 180

⟹ \angle Z\ = 180 -108

⟹ \angle Z\ = 72 degree

**Hence 72 degree is the right answer**