Same side interior angles

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.

Intersection of roads

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.

examples of corresponding angles

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."

Interior angles of parallel lines intersected by a transversal

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:

Co-interior angles

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.

 
important notes to remember
Important Notes
  1. The same side interior angles are non-adjacent and lie on the same side of the transversal.
  2. 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?

Same side interior angles example

Solution

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

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

Thus, 125o and 60o 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 \(\angle O P Q\).

Co-interior angles example

Solution

We will extend the lines in the given figure.

Co-interior angles example problem

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 \(x\).

Same side interior theorem examples: Two angles (2x+4) and (12x+8) are same side interior angles

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, \[ \begin{align} (2x+4)+(12x+8) &=180\\[0.2cm] 14x +12&=180\\[0.2cm] 14x&=168\\[0.2cm] x&=12 \end{align}\]
\(\therefore\) \(x=12\)
 
Challenge your math skills
Challenging Questions
  1. In the following figure, \(\mathrm{AB}\|\mathrm{CD}\| \mathrm{EF}\):

Challenging question on same side interior angles
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.

Interior angles of parallel lines intersected by a transversal

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

  • 1 and 3
  • 2 and 4
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