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)

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

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. 

Parallel sides	Transversal	Property
AB ∥ CD	BC	∠B + ∠C = 180°
AB ∥ CD	AD	∠A + ∠D = 180°
AD ∥ BC	CD	∠C + ∠D = 180°
AD ∥ BC	AB	∠A + ∠B = 180°

Facts about Same Side 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

Frequently Asked Questions about Same Side Interior Angles