# Same Side Interior Angles

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

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