Updated date:

Nov 18, 2020

Same-Side Interior Angles: Theorem, Proof, and Examples

Author:

Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.

The same-side interior angles are two angles that are on the same side of the transversal line and in between two intersected parallel lines. A transversal line is a straight line that intersects one or more lines.

The Same-Side Interior Angles Theorem states that if a transversal cuts two parallel lines, then the interior angles on the same side of the transversal are supplementary. Supplementary angles are ones that have a sum of 180°.

Same-Side Interior Angles Theorem Proof

Let L₁ and L₂ be parallel lines cut by a transversal T such that ∠2 and ∠3 in the figure below are interior angles on the same side of T. Let us show that ∠2 and ∠3 are supplementary.

Since ∠1 and ∠2 form a linear pair, then they are supplementary. That is, ∠1 + ∠2 = 180°. By the Alternate Interior Angle Theorem, ∠1 = ∠3. Thus, ∠3 + ∠2 = 180°. Therefore, ∠2 and ∠3 are supplementary.

The Converse of Same-Side Interior Angles Theorem

If a transversal cuts two lines and a pair of interior angles on the same side of the transversal is supplementary, then the lines are parallel.

The Converse of Same-Side Interior Angles Theorem Proof

Let L₁ and L₂ be two lines cut by transversal T such that ∠2 and ∠4 are supplementary, as shown in the figure. Let us prove that L₁ and L₂ are parallel.

Since ∠2 and ∠4 are supplementary, then ∠2 + ∠4 = 180°. By the definition of a linear pair, ∠1 and ∠4 form a linear pair. Thus, ∠1 + ∠4 = 180°. Using the transitive property, we have ∠2 + ∠4 = ∠1 + ∠4. By the addition property, ∠2 = ∠1

Hence, L₁ is parallel to L₂.

The Converse of Same-Side Interior Angles Theorem
John Ray Cuevas

Example 1: Finding the Angle Measures Using Same-Side Interior Angles Theorem

In the accompanying figure, segment AB and segment CD, ∠D = 104°, and ray AK bisect ∠DAB. Find the measure of ∠DAB, ∠DAK, and ∠KAB.

Solution

Since side AB and CD are parallel, then the interior angles, ∠D and ∠DAB, are supplementary. Thus, ∠DAB = 180° - 104° = 76°. Also, since ray AK bisects ∠DAB, then ∠DAK ≡ ∠KAB.

Final Answer

Therefore, ∠DAK = ∠KAB = (½)(76) = 38.

Example 2: Determining if Two Lines Cut by Transversal Are Parallel

Identify if lines A and B are parallel given the same-side interior angles, as shown in the figure below.

Example 3: Finding the Value of X of Two Same-Side Interior Angles

Find the value of x that will make L₁ and L₂ parallel.

Solution

The given equations are the same-side interior angles. Since the lines are considered parallel, the angles’ sum must be 180°. Make an expression that adds the two equations to 180°.

(3x + 45) + (2x + 40) = 180

5x + 85 = 180

5x = 180 – 85

5x = 95

x = 19

Final Answer

The final value of x that will satisfy the equation is 19.

Example 4: Finding the Value of X Given Equations of the Same-Side Interior Angles

Find the value of x given m∠4 = (3x + 6)° and m∠6 = (5x + 12)°.

Solution

The given equations are the same-side interior angles. Since the lines are considered parallel, the angles’ sum must be 180°. Make an expression that adds the expressions of m∠4 and m∠6 to 180°.

m∠4 + m∠4 = 180

3x + 6 + 5x + 12 = 180

8x + 20 = 180

8x = 180 - 20

8x = 160

x = 20

Final Answer

The final value of x that will satisfy the equation is 20.

Example 5: Finding the Value of Variable Y Using Same-Side Interior Angles Theorem

Solve for the value of y given its angle measure is the same-side interior angle with the 105° angle.

Solution

See to it that y and the obtuse angle 105° are same-side interior angles. It simply means that these two must equate to 180° to satisfy the Same-Side Interior Angles Theorem.

y + 105 = 180

y = 180 – 105

y = 75

Final Answer

The final value of x that will satisfy the theorem is 75.

Example 6: Finding the Angle Measure of All Same-Side Interior Angles

The lines L₁ and L₂ in the diagram shown below are parallel. Find the angle measures of m∠3, m∠4, and m∠5.

Solution

The lines L₁ and L₂ are parallel, and according to the Same-Side Interior Angles Theorem, angles on the same side must be supplementary. Note that m∠5 is supplementary to the given angle measure 62°, and

m∠5 + 62 = 180

m∠5 = 180 – 62

m∠5 = 118

Since m∠5 and m∠3 are supplementary. Make an expression adding the obtained angle measure of m∠5 with m∠3 to 180.

m∠5 + m∠3 = 180

118 + m∠3 = 180

m∠3 = 180 – 118

m∠3 = 62

The same concept goes for the angle measure m∠4 and the given angle 62°. Equate the sum of the two to 180.

62 + m∠4 = 180

m∠4 = 180 – 62

m∠4 = 118

It also shows that m∠5 and m∠4 are angles with the same angle measure.

Final Answer

m∠5 = 118°, m∠3 = 62°, m∠4 = 118°

Example 7: Proving Two Lines Are Not Parallel

The lines L₁ and L₂, as shown in the picture below, are not parallel. Describe the angle measure of z?

Solution

Given that L₁ and L₂ are not parallel, it is not allowed to assume that angles z and 58° are supplementary. The value of z cannot be 180° - 58° = 122°, but it could be any other measure of higher or lower measure. Also, it is evident with the diagram shown that L₁ and L₂ are not parallel. From there, it is easy to make a smart guess.

Final Answer

The angle measure of z = 122°, which implies that L₁ and L₂ are not parallel.

Example 8: Solving for the Angle Measures of Same-Side Interior Angles

Find the angle measures of ∠b, ∠c, ∠f, and ∠g using the Same-Side Interior Angle Theorem, given that the lines L₁, L₂, and L₃ are parallel.

Solution

Given that L₁ and L₂ are parallel, m∠b and 53° are supplementary. Create an algebraic equation showing that the sum of m∠b and 53° is 180°.

m∠b + 53 = 180

m∠b = 180 – 53

m∠b = 127

Since the transversal line cuts L₂, therefore m∠b and m ∠c are supplementary. Make an algebraic expression showing that the sum of ∠b and ∠c is 180°. Substitute the value of m∠b obtained earlier.

m∠b + m∠c = 180

127 + m∠c = 180

m∠c = 180 – 127

m∠c = 53

Since the lines L₁, L₂, and L₃ are parallel, and a straight transversal line cuts them, all the same-side interior angles between the lines L₁ and L₂ are the same with the same-side interior of L₂ and L₃.

m∠f = m∠b

m∠f = 127

m∠g = m∠c

m∠g = 53

Final Answer

m∠b = 127°, m∠c = 53°, m∠f = 127°, m∠g = 53°

Example 9: Identifying the Same-Side Interior Angles in a Diagram

Give the complex figure below; identify three same-side interior angles.

Solution

There are a lot of same-side interior angles present in the figure. Through keen observation, it is safe to infer that three out of many same-side interior angles are ∠6 and ∠10, ∠7 and ∠11, and ∠5 and ∠9.

Example 10: Determining Which Lines Are Parallel Given a Condition

Given ∠AFD and ∠BDF are supplementary, determine which lines in the figure are parallel.

Solution

By keen observation, given the condition that ∠AFD and ∠BDF are supplementary, the parallel lines are line AFJM and line BDI.

Explore Other Math Articles

How to Find the General Term of Sequences
This is a full guide in finding the general term of sequences. There are examples provided to show you the step-by-step procedure in finding the general term of a sequence.
Age and Mixture Problems and Solutions in Algebra
Age and mixture problems are tricky questions in Algebra. It requires deep analytical thinking skills and great knowledge in creating mathematical equations. Practice these age and mixture problems with solutions in Algebra.
AC Method: Factoring Quadratic Trinomials Using the AC Method
Find out how to perform AC method in determining if a trinomial is factorable. Once proven factorable, proceed with finding the factors of the trinomial using a 2 x 2 grid.
How to Solve for the Moment of Inertia of Irregular or Compound Shapes
This is a complete guide in solving for the moment of inertia of compound or irregular shapes. Know the basic steps and formulas needed and master solving moment of inertia.
Calculator Techniques for Quadrilaterals in Plane Geometry
Learn how to solve problems involving Quadrilaterals in Plane Geometry. It contains formulas, calculator techniques, descriptions, and properties needed in order to interpret and solve Quadrilateral problems.
How to Graph an Ellipse Given an Equation
Learn how to graph an ellipse given the general form and standard form. Know the different elements, properties, and formulas necessary in solving problems about ellipse.
How to Calculate the Approximate Area of Irregular Shapes Using Simpson’s 1/3 Rule
Learn how to approximate the area of irregularly shaped curve figures using Simpson’s 1/3 Rule. This article covers concepts, problems, and solutions about how to use Simpson’s 1/3 Rule in area approximation.
Finding the Surface Area and Volume of Frustums of a Pyramid and Cone
Learn how to calculate the surface area and volume of the frustums of the right circular cone and pyramid. This article talks about the concepts and formulas needed in solving for the surface area and volume of frustums of solids.
Finding the Surface Area and Volume of Truncated Cylinders and Prisms
Learn how to compute for the surface area and volume of truncated solids. This article covers concepts, formulas, problems, and solutions about truncated cylinders and prisms.
How to Use Descartes' Rule of Signs (With Examples)
Learn to use Descartes' Rule of Signs in determining the number of positive and negative zeros of a polynomial equation. This article is a full guide that defines Descartes' Rule of Signs, the procedure on how to use it, and detailed examples and sol
Solving Related Rates Problems in Calculus
Learn to solve different kinds of related rates problems in Calculus. This article is a full guide that shows the step-by-step procedure of solving problems involving related/associated rates.

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.