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

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.

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.

Solution

Apply the Same-Side Interior Angles Theorem in finding out if line A is parallel to line B. The theorem states that the same-side interior angles must be supplementary given the lines intersected by the transversal line are parallel. If the two angles add up to 180°, then line A is parallel to line B.

127° + 75° = 202°

Final Answer

Since the sum of the two interior angles is 202°, therefore the lines are not parallel.

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.

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© 2020 Ray