# Same side interior angles

Interior angles are fun to play around with once you know what exactly they are, and how to calculate them. Understanding interesting properties like the same side interior angles theorem and alternate interior angles help a long way in making the subject easier to understand.

Here is what happened with Ujjwal the other day.

Ujjwal was going in a car with his dad for a basketball practice session. On the way to the ground, he saw many roads intersecting the main road at multiple angles.

That's when his curiosity grew as to what is the relation between the angles created by the roads.

Let’s find out!

**Lesson Plan**

**What Are Same Side Interior Angles?**

When two parallel lines are intersected by a transversal, 8 angles are formed.

We will study more about "Same Side Interior Angles" here.

The same side interior angles are the pair of non-adjacent interior angles that lie on the same side of the transversal.

So the same side interior angles:

- have different vertices
- lie between two lines
- and are on the same side of the transversal

The "same side interior angles" are also known as "co-interior angles."

These 8 angles are classified into three types:

- Alternate Interior Angles
- Corresponding Angles
- Alternate Exterior Angles
- Same Side Interior Angles

In the above figure, \(L_1\) and \(L_2\) are parallel and \(L\) is the transversal.

From the "Same Side Interior Angles - Definition," the pairs of same side interior angles in the above figure are:

- 1 and 4
- 2 and 3

**Same Side Interior Angle Theorem**

The relation between the same side interior angles is determined by the same side interior angle theorem.

The "same side interior angle theorem" states:

If a transversal intersects two parallel lines, each pair of same side interior angles are supplementary (their sum is 180\(^\circ\)).

Conversely, if a transversal intersects two lines such that a pair of same side interior angles are supplementary, then the two lines are parallel.

We can see the "Same Side Interior Angle Theorem - Proof" and "Converse of Same Side Interior Angle Theorem - Proof" in the following sections.

**Same Side Interior Angle Theorem - Proof**

Refer to the following figure once again:

We have:

\[ \begin{align} \angle 1& = \angle 5 \;\;\;\text{ (corresponding angles)} \\[0.3cm]\angle 5 + \angle4& = 180^\circ \;\text{ (linear pair)}\end{align} \]

From the above two equations, \[\angle 1 + \angle4 = 180^\circ\]

Similarly, we can show that \[\angle 2 + \angle 3 = 180^\circ \]

**Converse of Same Side Interior Angle Theorem - Proof**

Let us assume that

\[ \begin{align}\angle 1 + \angle4 &= 180^\circ & \rightarrow (1) \end{align}\]

Since \(\angle 5\) and \(\angle 4\) forms linear pair,

\[ \begin{align}\angle 5 + \angle4 &= 180^\circ & \rightarrow (2) \end{align}\]

From (1) and (2),

\[ \angle 1 = \angle 5\]

Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.

Hence, the same side interior angle theorem is proved.

Would you like to observe visually how the same side interior angles are supplementary?

Here is an illustration for you to try.

Click on "Go" to see how the "Same Side Interior Angles Theorem" is true.

- The same side interior angles are non-adjacent and lie on the same side of the transversal.
- Two lines are parallel if and only if the same side interior angles are supplementary.

Our Math Experts are curating the same side interior angles worksheets for your child to practice the concept even when offline. For now, go through the Solved examples and the interactive questions that follow.

**Solved Examples**

Example 1 |

Are the following lines \(l\) and \(m\) parallel?

**Solution**

In the given figure, 125^{o} and 60^{o} are the same side interior angles if they are supplementary.

But \[125^\circ+60^\circ = 185^\circ\]

Thus, 125^{o} and 60^{o} are NOT supplementary.

Thus, by the "Same Side Interior Angle Theorem", the given lines are NOT parallel.

\(\therefore\) \(l\) and \(m\) are NOT parallel |

Example 2 |

In the following figure, \(M N \| O P\) and \(O N \| P Q\).

If \(\angle M N O=55^\circ\) then find \(\angle O P Q\).

**Solution**

We will extend the lines in the given figure.

Here, \(M N \| O P\) and \(ON\) is a transversal.

Thus, \(55^\circ\) and \(x\) are same side interior angles and hence, they are supplementary (by same side interior angle theorem). i.e.,

\[ \begin{align}55^\circ+x&=180^\circ\\[0.3cm] x &=125^\circ \end{align}\]

Again, \(O N \| P Q\) and \(OP\) is a transversal.

Thus, \(x\) and \(\angle O P Q\) are corresponding angles and hence they are equal. i.e.,

\[ \angle O P Q = x = 125^\circ\]

\(\therefore\) \(\angle O P Q=125^\circ\) |

Example 3 |

In the following figure, \(l\| m\).

Then find the value of \(x\).

**Solution**

Since \(l \| m\) and \(t\) is a transversal, \((2x+4)^\circ\) and \((12x+8)^\circ\) are same side interior angles.

Thus, by the "same side interior angle theorem", these angles are supplementary.

Thus, \[ \begin{align} (2x+4)+(12x+8) &=180\\[0.2cm] 14x +12&=180\\[0.2cm] 14x&=168\\[0.2cm] x&=12 \end{align}\]\(\therefore\) \(x=12\) |

- In the following figure, \(\mathrm{AB}\|\mathrm{CD}\| \mathrm{EF}\):

Find the value of \(x\)?

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

**Let's Summarize**

The mini-lesson targeted the fascinating concept of Same Side Interior Angles. The math journey around Same Side Interior Angles starts with what a student already knows, and goes on to creatively crafting a fresh concept in the young minds. Done in a way that not only it is relatable and easy to grasp, but also will stay with them forever. Here lies the magic with Cuemath.

**About Cuemath**

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we, at Cuemath, believe in.

**Frequently Asked Questions (FAQs)**

## 1. Why are same side interior angles congruent?

The same side interior angles are NOT congruent.

They are supplementary.

## 2. What do alternate interior angles mean?

Alternate interior angles are the pair of non-adjacent interior angles that lie on the opposite sides of the transversal.

## 3. Are same side interior angles adjacent?

The same side interior angles are always non-adjacent.

## 4. How do alternate interior angles look like?

Alternate interior angles are the pair of non-adjacent interior angles that lie on the opposite sides of the transversal.

In the above figure, the pairs of alternate interior angles are:

- 1 and 3
- 2 and 4