Learn what are consecutive interior angles with definition and diagram. Understand the consecutive interior angles theorem with its applications through examples.
Updated: 03/07/2022
Consecutive Interior Angles
What Are Consecutive Interior Angles?
Angles are created when two or more lines intersect, depending on how these lines cross, they will create different types of angles. In geometry, a pair of angles can relate to each other in a variety of ways. This helps identify them and use the theorem to solve problems related to angles. This lesson will focus on consecutive interior angles.
Consecutive interior angles are a pair of angles formed when a line, known as the transversal line, crosses two lines, parallel or non-parallel. The angles created on the inside of the two lines and are on the same side of the transversal line are known as the consecutive interior angles or co-interior angles for short.
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Looking at the image closely, one can identify the following:
The lines AB and CD are parallel lines
The line LM is a transversal line as it crossed lines AB and CD
{eq}\angle 3 {/eq} and {eq}\angle 5 {/eq} are co-interior angles
{eq}\angle 4 {/eq} and {eq}\angle 6 {/eq} are also co-interior angles
{eq}\angle 1 {/eq} and {eq}\angle 7 {/eq}\ are co-exterior angles
{eq}\angle 2 {/eq} and {eq}\angle 8 {/eq}\ are also co-exterior angles
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Consecutive Interior Angles Theorem
The consecutive interior angle theorem states that if a transversal line intersects a pair of parallel lines, then the sum of the consecutive interior angles is equal to 180 degrees. Using the sample illustration to determine the location of the angles, one can see that:
{eq}\angle 3 {/eq} + {eq}\angle 5 {/eq} = 180 degrees
{eq}\angle 4 {/eq}+ {eq}\angle 6 {/eq} = 180 degrees
In geometry, it is essential to prove the theorems, to better understand them, hence the following is a simple way to prove the consecutive interior angle theorem:
Given that {eq}\overline{AB} {/eq} and {eq}\overline{CD} {/eq} are parallel lines and {eq}\overline{LM} {/eq} is a transversal line. We know that
{eq}\angle 2 + \angle 4 {/eq} = 180 degrees (because they form a linear pair)
{eq}\angle 2 = \angle 6 {/eq}(because of the corresponding angle theorem)
If we substitute {eq}\angle 2 {/eq}with {eq}\angle 6 {/eq} we will get
{eq}\angle 6 + \angle 4 {/eq} = 180 degrees and therefore proving the consecutive interior angles theorem.
Converse of Consecutive Interior Angles Theorem
The converse of consecutive interior angles theorem states that if two lines are crossed by a transversal line and the consecutive interior angles are supplementary, which means when added they equal 180o, then the two lines, that were crossed, are parallel to one another.
The converse of consecutive interior angles theorem is used to identify if the lines that are crossed by a transversal line are parallel to each other or not.
Are Consecutive Interior Angles Congruent?
Two angles are considered to be congruent if they have the same measurement or degree to one another. Consecutive interior angles are not congruent, but they are supplemental angles. Meaning the sum of the consecutive interior angles is equal to 180 degrees when a transversal line crosses through two parallel lines.
It is important to note that this might not be true if a transversal line crosses through two non-parallel lines.
Supplementary Consecutive Interior Angles
Supplementary angles are two angles that when added together will equal 180o. It's important to note that not all consecutive interior angles are supplementary, meaning it doesn't always have to equal to 180 degrees. Consecutive interior angles are considered supplementary only when a transversal line crosses two parallel lines.
If a transversal line cross two non-parallel lines, then the consecutive interior angles formed will not be supplemental, meaning their sum will not equal to 180 degress.
Consecutive Interior Angles Examples
Now that you are familiar with the terminology used to describe and identify consecutive angles, here are some examples of when the consecutive interior angles theorem can be used:
Example 1: Howard drew a transversal line crossing two parallel lines on a piece of paper. He used a protractor to measure one of the consecutive interior angles, it equal 38 degrees, but before he could measure the other angles, his sister threw a ball and accidentally broke his protractor. Using the consecutive interior theorem, help Howard identify the measurement of the other consecutive interior angle?
Since the transversal line crossed a pair of parallel lines, the consecutive interior angles theorem can be used. This means that when we add the two consecutive interior angles their sum is equal to 180 degrees.
Therefore, subtracting 38 degrees from 180 degrees will give the solution to the missing angle.
180 - 38 = 142
Hence, the missing angle is equal to 142 degrees.
Example 2: Using the image provided, I identify the measurement of {eq}\angle 3 {/eq}.
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From the image, it is evident that {eq}\angle 3 {/eq} and {eq}\angle 5 {/eq} are consecutive interior angles, and since the lines that the transversal line crossed are parallel, the consecutive interior angles theorem can be applied to solve this problem.
{eq}\angle 5 {/eq} = 115 degrees, subtracting that from 180 degrees will give the solution.
180 - 115 = 65
Therefore, {eq}\angle 3 {/eq} is equal to 65 degrees.
Example 3: Using the image provided, I identify the measurement of {eq}\angle 6 {/eq}
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From the image, it is evident that {eq}\angle 4 {/eq} and {eq}\angle 6 {/eq} are consecutive interior angles, and since the lines that the transversal line crossed are parallel, the consecutive interior angles theorem can be applied to solve this problem.
{eq}\angle 4 {/eq}= 97 degrees, subtracting that from 180o will give the solution.
180 - 97 = 83
Therefore, {eq}\angle 6 {/eq} is equal to 83 degrees.
Lesson Summary
Consecutive interior angles are a pair of angles formed when a line, known as the transversal line, crosses two lines. The angles created on the inside of the two lines and are on the same side of the transversal line are known as the consecutive interior angles or co-interior angles for short.
The consecutive interior angle theorem states that if a transversal line intersects a pair of parallel lines, then the sum of the consecutive interior angles is equal to 180 degrees forming what is known as supplemental angles.
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Frequently Asked Questions
Are consecutive interior angles congruent?
Consecutive Interior Angles are not congruent, they are supplemental and add up to 180 degrees if the two lines are crossed by the transversal line as parallel to one another.
What is the consecutive exterior angles theorem?
The consecutive interior angle theorem states that if a transversal line intersects a pair of parallel lines, then the sum of the consecutive interior angles is equal to 180o.
How do you know if angles are consecutive interior angles?
Two angles are consecutive interior angles if they follow these 2 properties. The two angles are both on the same side of a transversal AND they are both inside the two lines being crossed.
Do consecutive interior angles add up to 180?
Consecutive Interior Angles add up to 180 degrees, only if the two lines being crossed by a transversal line are parallel to one another.
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