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Consecutive Interior Angles Theorem

When we cut two lines with a third line, we''re left with a number of new angles. Thanks to the consecutive interior angles theorem, we can immediately make a number of deductions about these angles. But how exactly does this theorem work? How is it different compared to other theorems? What can it teach us about math? Let''s find out:

The consecutive interior angles theorem explained

Imagine that we have two lines. What happens when we cut through both lines with another, third line?

In this situation, we create a total of eight angles in the areas where these lines intersect.

Among these angles are consecutive interior angles. These are the angles that occur in the interior of the transversal and on the same side. Remember that a transversal is simply a line that cuts through other lines.

The interior angles theorem states that these consecutive interior angles are supplementary when the two transverse lines are parallel. In other words, they add up to exactly 180 degrees (a straight line).

Visualizing the consecutive interior angles theorem

Take a closer look at how the consecutive interior angles theorem works:

In this diagram, we can see that two pairs of consecutive interior angles have been highlighted. One pair -- angles 3 and 5 -- has been highlighted in red. Another pair -- angles 4 and 6 -- have been highlighted in blue. Note that lines a and b are not parallel, so the consecutive interior angles theorem does not apply in this situation.

Proving that the consecutive interior angles theorem is valid

Proving that the consecutive interior angles theorem is valid

Let''s start with our goal: Prove that if two parallel lines are cut by a transversal, then consecutive interior angles (angles 3 and 5 and angles 4 and 6 in this case) are supplementary pairs.

First, we assume that the two lines being cut by the transversal are parallel.

Next, we state that angles 1 and 3 and angles 2 and 4 form two linear pairs, respectively.

If angles 1 and 3 are supplementary, we can use the supplementary angles postulate to determine that they equal 180 degrees. We can say the same for angles 2 and 4.

Since the lines are parallel and cut by a transversal, we can apply the corresponding angles postulate: angle 1 is congruent to angle 5, and angle 2 is congruent to angle 6.

Finally, we know by the substitution property that angles 3 and 5 are supplementary, and angles 4 and 6 are also supplementary.

Topics related to the Consecutive Interior Angles Theorem

Alternate Interior Angles Theorem

Polygon Interior Angles Sum Theorem

Exterior Angle Theorem

Flashcards covering the Consecutive Interior Angles Theorem

Basic Geometry Flashcards

Common Core: High School - Geometry Flashcards

Practice tests covering the Consecutive Interior Angles Theorem

Common Core: High School - Geometry Diagnostic Tests

Intermediate Geometry Diagnostic Tests

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