# Same Side Interior Angles – Definition, Theorem, Examples, Facts Home » Math-Vocabluary » Same Side Interior Angles – Definition, Theorem, Examples, Facts

## What Are the Same Side Interior Angles in Geometry?

Same side interior angles are a pair of non-adjacent angles formed by two parallel lines (or non-parallel lines) cut by a transversal. They lie on the same side of the transversal and in the interior region between two lines.

The same side interior angles are also called co-interior angles or consecutive interior angles.

The given diagram shows the same side interior angles formed when i) parallel lines are cut by a transversal ii) two non-parallel lines are cut by a transversal

## Same Side Interior Angles: Definition

Two non-adjacent interior angles that lie on the same side of the transversal are known as Same Side Interior Angles.

## How To Spot or Identify Same Side Interior Angles

Same side interior angles form a C-shaped pattern due to their position.

## Properties of Same Side Interior Angles

• They have different vertices.
• They lie on the same side of the transversal.
• They lie in the interior region formed between the two lines.
• They share a common side.

## Same Side Interior Angle Theorem (Same Side Interior Angles Postulate)

Same Side Interior Angle Theorem Statement: If a transversal intersects two parallel lines, then each pair of same side interior angles are supplementary (they add up to 180°.)

## Proof of Same Side Interior Angles Theorem

Given: l || m are parallel, and n is the transversal.

To prove:

∠4 + ∠5 = 180°

∠3 + ∠6 = 180°

Proof:

∠1 = ∠5 (corresponding angles)

∠1 + ∠4 = 180° (Linear pair of angles)

On Substituting ∠1 = ∠5, we get

∠4 + ∠5 = 180°

Similarly, we can show that

∠3 +∠6 = 180°

Thus, the same side interior angles are supplementary.

## Converse of the Same Side Interior Angles Theorem

Converse of Same Side Interior Angle Theorem: If a transversal intersects two lines in a way that a pair of the same side interior angles are supplementary, then the two lines are said to be parallel.

Given:

∠3 +∠6 = 180°

∠4 + ∠5 = 180°

To prove: l || m

∠5 + ∠4 = 180° (given)

∠1 + ∠4 = 180° (linear pair of angles)

On comparing, we get

∠1 + ∠4 = ∠5 + ∠4

∠1 = ∠5

∠1 and ∠5 are a pair of corresponding angles.

l is parallel to m. (Converse of corresponding angles postulate)

## Same Side Interior Angles of a Parallelogram

The opposite sides of a parallelogram are parallel. So, any two adjacent angles of a parallelogram are basically the same side interior angles formed by two parallel lines and a transversal.

We know that same side interior angles are supplementary when lines are parallel. This gives rise to the important property in parallelograms which states that:

Two adjacent angles of a parallelogram are always supplementary.

In the following parallelogram, ∠A and ∠B, ∠B and ∠C, ∠C and ∠D, and ∠D and ∠A are pairs of consecutive interior angles.

## Facts about Same Side Interior Angles

• Same-side interior angles are supplementary when the lines intersected by the transversal line are parallel.
• Same side interior angles are also known as consecutive interior angles or co-interior angles.

## Conclusion

In this article, we learned about same side interior angles, which are also known as co-interior angles. We learned the important theorem based on same side interior angles that helps to determine whether two lines are parallel or not. Let’s solve a few examples for better understanding.

## Solved Examples on Same Side Interior Angles

1. Is l parallel to m?

Solution:

The given pair of angles are the same side interior angles.

120° + 60° = 180°

The angles are supplementary.

Hence, l is parallel to m.

2. In the following figure, find the value of x.

Solution:

Two parallels are cut by a transversal.

Given angles represent a pair of same side interior angles.

They must add up to 180°.

So, 141° + x = 180° (Same Side Interior Angles)

x = 180° – 141°

x = 39°

3. Find the value of x in the following figure,

Solution:

2x + 3 + x – 12 = 180° (Same Side Interior Angles)

3x – 9 = 180°

3x = 180° + 9°

3x = 189°

$\Rightarrow x = \frac{189^{\circ}}{3} = 63^{\circ}$

4. Two parallel lines are cut by a transversal, and a pair of same side interior angles measure (3y + 6)° and (9y – 24)° respectively. Find the value of y.

Solution:

(3y + 6)° + (9y – 24)° = 180° (Same Side Interior Angles)

12y – 18° = 180°

12y = 180° + 18°

12y = 198°

y $= \frac{198^{\circ}}{12}$

y = 16.5°

## Practice Problems on Same Side Interior Angles

1

### Which of the following is the pair of Same Side Interior Angles? ∠3 and ∠4
∠1 and ∠3
∠3 and ∠2
∠1 and ∠4
CorrectIncorrect
Correct answer is: ∠3 and ∠4
Consecutive Interior Angles are the angles which are on the same side of transversal and are on the interior part. So, ∠3 and ∠4 are same side interior angles.
2

### Which of the following is true for the following figure? P is perpendicular to Q.
P and Q intersect each other.
P is parallel to Q.
P is not parallel to Q.
CorrectIncorrect
Correct answer is: P is not parallel to Q.
150° + 20° = 170°
Angles are not supplementary.
Hence, line P is not parallel to line Q.
3

### If one of the same side interior angles is double the other, then what will be the value of each of the angles?

40° and 160°
80° and 100°
45° and 90°
60° and 120°
CorrectIncorrect
Correct answer is: 60° and 120°
Let one consecutive interior angle be x°.
Other interior angle = 2x°
x° + 2x° = 180°
3x° = 180°
x = 60°
2x = 2 60° = 120°
4

### For which value is each of the same side interior angles equal?

60°
80°
90°
100°
CorrectIncorrect
Let each consecutive interior angle be x.
x + x = 180°
2x = 180°
x $= \frac{180^{\circ}}{2} = 90^{\circ}$
5

### What is the value of each angle in the following figure? 92° and 88°
100° and 80°
75° and 105°
95° and 85°
CorrectIncorrect
Correct answer is: 95° and 85°
m is parallel to n and p is the transversal.
5x° + (3x+28)° = 180° (Same Side Interior Angles)
8x + 28° = 180°
8x = 180° – 28°
8x = 152°
x $= \frac{152^{\circ}}{8} = 19^{\circ}$
Angles = 5 19° = 95° and 3 19° + 28° = 57° + 28° = 85°