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# 2.6: Parallel Lines and Transversals: Identifying Angle Pairs, Part 2

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

• Identify angles formed by two parallel lines and a non-perpendicular transversal.
• Identify and use the Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, and Same-Side Interior Angles Theorem.

## Parallel Lines with a Transversal — Review of Terms

As a quick review, it is helpful to practice identifying different categories of angles.

Example 1

In the diagram below, two vertical parallel lines are cut by a transversal. Identify the pairs of corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.

Corresponding angles: 5\begin{align*}\angle 5\end{align*} and 3\begin{align*}\angle 3\end{align*}, ____________, ____________, and ____________.

• Corresponding angles are formed on different lines, but in the same relative position to the transversal — in other words, they face the same direction.
• There are four pairs of corresponding angles in this diagram — 6\begin{align*}\angle 6\end{align*} and 8,7\begin{align*}\angle 8, \angle 7\end{align*} and 1,5\begin{align*}\angle 1, \angle 5\end{align*} and 3,\begin{align*}\angle 3,\end{align*} and 4\begin{align*}\angle 4\end{align*} and 2\begin{align*}\angle 2\end{align*}.

(Using the same diagram on the previous page):

Alternate interior angles: 7\begin{align*}\angle 7\end{align*} and 3\begin{align*}\angle 3\end{align*}, and ____________.

• These angles are on the interior of the lines crossed by the transversal and are on opposite sides of the transversal.
• There are two pairs of alternate interior angles in this diagram — 7\begin{align*}\angle 7\end{align*} and 3,\begin{align*}\angle 3,\end{align*} and 8\begin{align*}\angle 8\end{align*} and 4\begin{align*}\angle 4\end{align*}.

Alternate exterior angles: 6\begin{align*}\angle 6\end{align*} and2,\begin{align*} \angle 2,\end{align*} and ____________.

• These are on the exterior of the lines crossed by the transversal and are on opposite sides of the transversal.
• There are two pairs of alternate exterior angles in this diagram — 6\begin{align*}\angle 6\end{align*} and 2,\begin{align*}\angle 2,\end{align*} and 5\begin{align*}\angle 5\end{align*} and 1\begin{align*}\angle 1\end{align*}.

Consecutive (same-side) interior angles: 7\begin{align*}\angle 7\end{align*} and 8,\begin{align*}\angle 8,\end{align*} and ____________.

• Consecutive interior angles are in the interior region of the lines crossed by the transversal, and are on the same side of the transversal.
• There are two pairs of consecutive interior angles in this diagram — 7\begin{align*}\angle 7\end{align*} and 8,\begin{align*}\angle 8,\end{align*} and 3\begin{align*}\angle 3\end{align*} and 4\begin{align*}\angle 4\end{align*}.

## Angle Postulates and Theorems

By now you have had lots of practice and should be able to identify relationships between angles.

Do you remember the difference between a postulate and a theorem?

The difference is...

A postulate does not have to be proven. It is self-evident, or obvious. A theorem is not obvious. It has to be proven.

A postulate does not have to be ______________________________.

A ______________________________ must be proven.

We will explore a number of postulates and theorems that involve the different types of angle relationships you just learned.

Corresponding Angles Postulate

If the lines crossed by a transversal are parallel, then corresponding angles will be congruent.

Examine the following diagram:

You already know that 2\begin{align*}\angle 2\end{align*} and 3\begin{align*}\angle 3\end{align*} are corresponding angles because they are formed by two lines crossed by a transversal and have the same relative placement next to the transversal.

The Corresponding Angles Postulate says that because the lines are parallel to each other (which we can tell because of the similar arrows on them), the corresponding angles will be congruent.

• We know the two lines in the diagram above are parallel because they are both marked with __________________________.
• The Corresponding Angles Postulate tells us that corresponding angles are __________________________________ if the transversal crosses parallel lines.

In the diagram below, lines p\begin{align*}p\end{align*} and q\begin{align*}q\end{align*} are parallel. What is the measure of 1\begin{align*}\angle 1\end{align*}?

Because lines p\begin{align*}p\end{align*} and q\begin{align*}q\end{align*} are parallel, the 120\begin{align*}120^\circ\end{align*} angle and 1\begin{align*}\angle 1\end{align*} are corresponding angles. We know by the Corresponding Angles Postulate that they are congruent. Therefore, m1=120\begin{align*}m \angle 1=120^\circ\end{align*}.

Alternate Interior Angles Theorem

You can use the Corresponding Angles Postulate to derive the relationships between all other angles formed when two lines are crossed by a transversal.

Look at the diagram below and the chart on the following page:

If you know that the measure of angle 1 is 120\begin{align*}120^\circ\end{align*}, you can find the measures of the other angles in the picture.

What are the measures of angles 2, 3, and 4?

Try to figure them out and then check your work in the chart on the next page.

Angle Measure How We Know
2=60\begin{align*}\angle 2 = 60^\circ\end{align*}

Because it makes a linear pair with 1\begin{align*}\angle 1\end{align*}.

Remember, linear pairs are supplementary. That means they add to be 180\begin{align*}180^\circ\end{align*}

120+60=180\begin{align*}120 + 60^\circ = 180^\circ\end{align*}

3=120\begin{align*}\angle 3 = 120^\circ\end{align*}

There are two ways to find the measure of 3\begin{align*}\angle 3\end{align*}. First, 3\begin{align*}\angle 3\end{align*} and 1\begin{align*}\angle 1\end{align*} are vertical angles, and vertical angles are congruent.

120=120\begin{align*}120^\circ = 120^\circ\end{align*}

Or, you can find the measure of Angle 3 using linear pairs.

2\begin{align*}\angle 2\end{align*} and 3\begin{align*}\angle 3\end{align*} make a linear pair. This means they are supplementary (add to 180\begin{align*}180^\circ\end{align*}).

We already know that 2=60\begin{align*}\angle 2 = 60^\circ\end{align*}, so \begin{align*}\angle 3\end{align*} must be \begin{align*}120^\circ\end{align*}.

\begin{align*}60^\circ + 120^\circ = 180^\circ\end{align*}

\begin{align*}\angle 4 = 60^\circ\end{align*}

There are two ways to find the measure of \begin{align*}\angle 4\end{align*}. First, \begin{align*}\angle 4\end{align*} and \begin{align*}\angle 2\end{align*} are vertical angles, and vertical angles are congruent. We know that \begin{align*}\angle 2 = 60^\circ\end{align*}

\begin{align*}60^\circ = 60^\circ\end{align*}

Or, you can find the measure of Angle 4 using linear pairs. \begin{align*}\angle 3\end{align*} and \begin{align*}\angle 4\end{align*} make a linear pair. So do \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*}. This means they are supplementary (add to \begin{align*}180^\circ\end{align*}).

We already know that \begin{align*}\angle 3 = 120^\circ\end{align*}, so \begin{align*}\angle 4\end{align*} must be \begin{align*}60^\circ\end{align*}.

\begin{align*}60^\circ + 120^\circ = 180^\circ\end{align*}

As you can see from this chart, there are some very important relationships we already know that can help us find the measure of missing angles! In this example, the ones we used multiple times are:

• Linear pairs are supplementary and
• Vertical angles are congruent

Example 2

Lines \begin{align*}l\end{align*} and \begin{align*}m\end{align*} in the diagram below are parallel. What are the measures of angles \begin{align*}\alpha\end{align*} and \begin{align*}\beta\end{align*}?

In this problem, you need to find the angle measures of two alternate interior angles given an exterior angle. Use what you know.

• There is one angle that measures \begin{align*}80^\circ\end{align*}. Angle \begin{align*}\beta\end{align*} and the \begin{align*}80^\circ\end{align*} angle are corresponding angles. So by the Corresponding Angles Postulate, \begin{align*}m \angle \beta = 80^\circ\end{align*}.
• Now, because \begin{align*}\angle \alpha\end{align*} is made by the same intersecting lines and is opposite the \begin{align*}80^\circ\end{align*} angle, these two angles are vertical angles. Since you already learned that vertical angles are congruent, we conclude \begin{align*}m \angle \alpha =80^\circ\end{align*}.
• Finally, compare angles \begin{align*}\alpha\end{align*} and \begin{align*}\beta\end{align*}. They both measure \begin{align*}80^\circ\end{align*}, so they are congruent. This will be true any time two parallel lines are cut by a transversal.

Alternate Interior Angles Theorem

Alternate interior angles formed by two parallel lines and a transversal are always congruent.

• When two parallel lines are crossed by a transversal, the alternate interior angles are _________________________________.

Alternate Exterior Angles Theorem

Now you know that pairs of...

• corresponding,
• vertical, and
• alternate interior angles are congruent.

We will use logic to show that Alternate Exterior Angles are congruent — when two parallel lines are crossed by a transversal, of course.

Example 3

Lines \begin{align*}g\end{align*} and \begin{align*}h\end{align*} in the diagram below are parallel. If \begin{align*}m \angle 4=43^\circ\end{align*}, what is the measure of \begin{align*}\angle 5\end{align*}?

You know from the problem that \begin{align*}m \angle 4=43^\circ\end{align*}. That means that \begin{align*}\angle 4\end{align*}’s corresponding angle, which is \begin{align*}\angle 3\end{align*}, will measure \begin{align*}43^\circ\end{align*} as well:

The corresponding angle you just filled in is also vertical to \begin{align*}\angle 5\end{align*}. Since vertical angles are congruent, you can conclude that \begin{align*}m \angle 5=43^\circ\end{align*}:

So, \begin{align*}\angle 4\end{align*} is congruent to \begin{align*}\angle 5\end{align*}. In other words, the alternate exterior angles are congruent.

Alternate Exterior Angles Theorem

If two parallel lines are crossed by a transversal, then alternate exterior angles are congruent.

In Example 3 on the previous page, we proved the Alternate Exterior Angles Theorem. We figured this out because:

• When lines are parallel, corresponding angles are ___________________________.
• Then, all vertical angles are _______________________________.

These led us to our conclusion that:

• When two parallel lines are cut by a transversal, the alternate exterior angles formed are _______________________________.

Consecutive (Same-Side) Interior Angles Theorem

The last category of angles to explore in this lesson is consecutive interior angles. They fall on the interior of the parallel lines and are on the same side of the transversal. Use your knowledge of corresponding angles to identify their mathematical relationship.

Example 4

Lines \begin{align*}r\end{align*} and \begin{align*}s\end{align*} in the diagram below are parallel.

If the angle corresponding to \begin{align*}\angle 1\end{align*} measures \begin{align*}76^\circ\end{align*}, what is \begin{align*}m \angle 2\end{align*}?

This process should now seem familiar. The given \begin{align*}76^\circ\end{align*} angle is adjacent to \begin{align*}\angle 2\end{align*} and they form a linear pair. Therefore, the angles are supplementary.

Since supplementary angles add up to __________, find \begin{align*}m \angle 2\end{align*} by subtracting \begin{align*}76^\circ\end{align*} from \begin{align*}180^\circ\end{align*}:

\begin{align*}m \angle 2=180^\circ-76^\circ\end{align*}

\begin{align*}m \angle 2=104^\circ\end{align*}

Example 4 on the last page shows that if two parallel lines are cut by a transversal, the consecutive interior angles are supplementary; they sum to \begin{align*}180^\circ\end{align*}.

This is called the Consecutive Interior Angles Theorem. We restate it below for clarity.

Consecutive (Same-Side) Interior Angles Theorem

If two parallel lines are crossed by a transversal, then consecutive interior angles are supplementary.

• Consecutive Interior Angles and Same-Side Interior Angles are the ______________.
• When two parallel lines are crossed by a transversal, same-side interior angles are _______________________________________.

In the diagram below, lines \begin{align*}m\end{align*} and \begin{align*}n\end{align*} are being cut by transversal \begin{align*}t\end{align*}.

For questions 1-3, circle the best answer.

1. Which postulate or theorem explains why angles \begin{align*}x\end{align*} and \begin{align*}q\end{align*} are congruent?

a. Corresponding Angles Postulate

b. Alternate Interior Angles Theorem

c. Alternate Exterior Angles Theorem

d. Consecutive (Same-Side) Interior Angles Theorem

2. If \begin{align*}m \angle w = 70^\circ\end{align*}, what will be the measure of \begin{align*}\angle s\end{align*}? What is the reason for your answer?

a. \begin{align*}70^\circ\end{align*}; Corresponding Angles Postulate

b. \begin{align*}70^\circ\end{align*}; Alternate Exterior Angles Theorem

c. \begin{align*}110^\circ\end{align*}; Corresponding Angles Postulate

d. \begin{align*}110^\circ\end{align*}; Alternate Exterior Angles Theorem

3. Which postulate or theorem states that an angle pair is supplementary (not congruent)?

a. Corresponding Angles Postulate

b. Alternate Interior Angles Theorem

c. Alternate Exterior Angles Theorem

d. Consecutive (Same-Side) Interior Angles Theorem

4. Label all of the angle measures in the diagram above.

## Graphic Organizer for Lesson 3

Parallel Lines and Angle Pairs
Angle Pairs Congruent or Supplementary? Picture
Corresponding
Alternate Interior
Alternate Exterior
Same-side/Consecutive Interior

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