When two lines are cut by a transversal and the same side interior angles are supplementary then the lines are parallel?

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Same side interior angles are two angles that are on the interior of (between) the two lines and specifically on the same side of the transversal. The same-side interior angles sum up to 180 degrees. When two parallel lines are intersected by a transversal line they formed 4 interior angles. The 2 non-adjacent interior angles that are on the same side of the transversal are supplementary.

What are Same Side Interior Angles?

When two parallel lines are intersected by a transversal, 8 angles are formed. 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 no common vertices or have different vertices
• lie between two lines
• and formed on the same side of the transversal

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

The 8 angles thus formed are classified into different types of angles listed below:

In the given figure, line AB || CD and line l is the transversal.

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

Same Side Interior Angles Theorem

Let us consider the above figure. In the above figure, lines AB and CD are parallel and L is the transversal. We just read that the pairs of the same side interior angles in the above figure are:

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

The theorem for 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°).

Same Side Interior Angles Theorem Proof

Referring the above figure once again:

∠4 = ∠8, and ∠3 = ∠7 [corresponding angles are equal].
∠5 + ∠8 = 180° and ∠6 + ∠7 = 180° [ linear pair of angles].

From the above two equations, ∠4 + ∠5 = 180°

Similarly, ∠3 + ∠6 = 180°

Hence proved, that each pair of same-side interior angles are supplementary.

Converse of Same Side Interior Angles Theorem

The converse of the same-side interior angle theorem states that if a transversal intersects two lines such that a pair of same-side interior angles are supplementary, then the two lines are parallel.

Converse of Same Side Interior Angles Theorem Proof

Considering same above figure,
Let us assume that

∠4 + ∠5 = 180° ⇒ (1)

Since ∠5 and ∠8 forms linear pair,

∠5 + ∠8 = 180° ⇒ (2)

From (1) and (2),

∠4 = ∠8

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

Hence, the converse of the same side interior angle theorem is proved.

Important Notes

The following are the important points related to the same side interior angles.

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

☛ Related Articles

Check out these interesting articles to know more about the same side interior angles and their related topics.

1. Example 1: In the given figure, 145° and 40° are the same side interior angles. Check whether the lines l and m are parallel or not.

   

   Solution: In the given figure, 145° and 40° are the same side interior angles.

   But the sum is not equal to 180° (145° + 40° =185°).

   Thus, 145° and 40° are NOT supplementary, their sum is not equal to 180°.

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

   Thus, lines l and m are not parallel.

2. Example 2: In the following figure, l || m and (4x + 4)° and (10x + 8)° are the same side interior angles. Find the value of x.

   

   Solution: Since l || m and t is a transversal, (4x + 4)° and (10x + 8)° are the same side interior angles.

   Thus, by the "same side interior angle theorem", these angles are supplementary or we can say that their sum is equal to 180°.

   Thus, (4x + 4) + (10x + 8) = 180

   14x + 12 = 180

   14x = 180 - 12

   14x = 168

   x = 12

   Thus, the value of x = 12.

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FAQs on Same Side Interior Angles

The same side interior angles are NOT congruent. They are supplementary. The same side interior angles formed when two parallel lines intersected by a transversal. The same side interior angles can be congruent only when each angle is equal to a 90 degree because then the sum of the same side interior angles is equal to 180 degrees.

Are Same Side Interior Angles Adjacent?

The same side interior angles are always non-adjacent because the angles are formed on the two different lines that are parallel to each other.

What is the Sum of the Two Same Side Interior Angles on the Transversal?

When two parallel lines crossed by a transversal they formed same-side interior angles and their sum is equal to 180 degrees. As the sum of the same side interior angles is 180 degrees therefore the angles are supplementary.

What is the Converse of Same Side Interior Angles?

The converse of the same-side interior angle states that when two lines intersected by a transversal and the angles inside on the same side are supplementary or we can say the sum of inside angles on the same side is 180 degrees then the lines are said to be parallel.

What is Another Name of the Same Side Interior Angles?

The same side interior angles are also known as consecutive interior angles as the angles are on one side of the transversal but inside the two parallel lines.

What is the Difference Between Same Side Interior Angles and Same Side Exterior Angles?

When two parallel lines are intersected by a transversal line 8 angles were formed. The same side interior angles are the angles inside the parallel lines on the same side of the transversal and the same side exterior angles are the angles outside the parallel lines on the same side of the transversal.

What is the Difference Between the Same Side Interior Angles and Corresponding Angles?

The difference between the same side interior angles and corresponding angles is corresponding angles are congruent whereas, in the case of the same side of interior angles, the sum of the same side interior angles is equal to 180 degrees only if the transversal line intersects two parallel lines.

Whenever two parallel lines are cut by a transversal, an interesting relationship exists between the two interior angles on the same side of the transversal. These two interior angles are supplementary angles. A similar claim can be made for the pair of exterior angles on the same side of the transversal. There are two theorems to state and prove. I'll give formal statements for both theorems, and write out the formal proof for the first. The second theorem will provide yet another opportunity for you to polish your formal proof writing skills.

• Theorem 10.4: If two parallel lines are cut by a transversal, then the interior angles on the same side of the transversal are supplementary angles.
• Theorem 10.5: If two parallel lines are cut by a transversal, then the exterior angles on the same side of the transversal are supplementary angles.

Let the fun begin. As promised, I will show you how to prove Theorem 10.4.

Figure 10.6 illustrates the ideas involved in proving this theorem. You have two parallel lines, l and m, cut by a transversal t. You will be focusing on interior angles on the same side of the transversal: 2 and 3. You'll need to relate to one of these angles using one of the following: corresponding angles, vertical angles, or alternate interior angles. There are many different approaches to this problem. Because Theorem 10.2 is fresh in your mind, I will work with 1 and 3, which together form a pair ofalternate interior angles.

Figure 10.6l m cut by a transversal t.

• Given: l m cut by a transversal t.
• Prove: 2 and 3 are supplementary angles.
• Proof: You will need to use the definition of supplementary angles, and you'll use Theorem 10.2: When two parallel lines are cut by a transversal, the alternate interior angles are congruent. That should be enough to complete the proof.

	Statements	Reasons
1.	l m cut by a transversal t	Given
2.	2 and 3 are same-side interior angles	Definition of same-side interior angles
3.	1 and 3 are alternate interior angles	Definition of alternate interior angles
4.	1 and 2 are supplementary angles, and m1 + m2 = 180º	Definition of supplementary angles
5.	1 ~= 3	Theorem 10.2
6.	m1 = m3	Definition of ~=
7.	m13 + m2 = 180º	Substitution (steps 4 and 6)
8.	2 and 3 are supplementary angles	Definition of supplementary angles

Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.

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