Alternate Interior Angles

Alternate Interior Angles (Definition, Theorem, & Examples)

1. Video
2. Key Terms
3. Alternate Interior Angles Definition
4. Alternate Interior Angles Theorem
• Converse Theorem
5. Alternate Interior Angles Examples
6. Alternate Interior Angles In Real Life

Key Terms

Sometimes geometry feels like a giant parts warehouse. You trade a lot of number-crunching (not much addition, multiplication, subtraction or division in geometry) for a lot of inventory.

Parts of an Angle

For example, let's construct $angle Z$. We almost never write "$angle Z$," using instead a quick shorthand, $\angle Z$. Something as simple as an angle has parts.

1. Two rays, $ZA$ and $ZU$, meet at $Point Z$
2. Where they meet at $Point Z$, they form a vertex, $\angle Z$

We say rays $ZA$ and $ZU$, but those rays could also be small snippets out of longer lines that intersected at $Point Z$. They could be snippets cut as rays or as line segments, depending on taking an infinite chunk or a finite chunk of the infinite, intersecting lines.

Parallel Lines

Unlike the intersecting rays $ZA$ and $ZU$, parallel lines never meet. The two lines, line segments, or rays never converge (move closer) or diverge (move away). The only sneaky way to get an angle from parallel lines is to declare each line is a straight angle, with a measure of $180°$. While two points determine a line, if you locate three points on a line, you have created a straight angle with the middle point as the vertex.

Transversals

Parallel lines can be intersected by transversals. Any line cutting across parallel lines is a transversal. It can cross at any angle. A transversal intersecting parallel lines at $90°$ is perpendicular.

Alternate Interior Angles Definition

When a transversal intersects parallel lines, it creates an interior and exterior. Think about it: if you were two-dimensional and came across a line in your path, that line would stretch infinitely in two directions and you could not get past it. You would be outside, at the exterior, of the parallel lines. Just beyond the line and between it and the parallel line next to it, is the interior.

When the transversal intersects, it creates four angles at each parallel line, or eight angles altogether. Four of those angles are exterior and four are interior. We are interested in the four interior lines, those are our Alternate Interior Angles.

Let's create parallel lines $LI$ and $ON$, and a transversal $HE$.

The two points where $HE$ crosses the parallel lines are $Points A$ and $R$. Yes, we have a $HARE$ crossing a $LION$.

You mark drawings of parallel lines with little bird-feet marks, like Vs on their sides. Notice we have four exterior angles:

1. $\angle HAL$
2. $\angle HAI$
3. $\angle ORE$
4. $\angle NRE$

We have four interior angles:

1. $\angle LAR$
2. $\angle IAR$
3. $\angle ARO$
4. $\angle ARN$

We are only interested in the four interior angles. Two of the interior angles are built using the top parallel line $LI$, and two are built using the bottom parallel line, $ON$.

Alternate interior angles are two congruent angles from different parallel lines (one from $LI$, one from $ON$). This says:

• $\angle LAR$ is an alternate interior angle with $\angle ARN$
• $\angle IAR$ is an alternate interior angle with $\angle ARO$.

Alternate Interior Angles Theorem

Once you can recognize and break apart the various parts of parallel lines with transversals, you can use the Alternate Interior Angles Theorem to speed up your work.

Converse of Alternate Interior Angles Theorem

The converse of the theorem is also true:

Alternate Interior Angles Examples

We can prove both these theorems so you can add them to your toolbox. With our origional figure, $LI$ and $ON$ are parallel lines (given) transversed by $HE$ (given).

We could declare all sorts of relationships, but the proof can be short and simple:

• $LI \parallel ON$ (given)
• $\angle LAR \cong \angle ORE$ (Corresponding Angles Postulate)
• $\angle ORE \cong \angle ARN$ (Vertical Angles Theorem)
• $\angle LAR \cong \angle ARN$ (Transitive Property of Congruence)

The Transitive Property of Congruence says if $A$ is like $B$ and $B$ is like $C$, then $A$ is like $C$. Since $\angle LAR$ was congruent to $\angle ORE$, and $\angle ORE$ was congruent to $\angle ARN$, then:

• $\angle LAR \cong \angle ORE \cong \angle ARN$
• $\angle LAR \cong \angle ARN$

To prove the converse, "If two lines are cut by a transversal and the alternate interior angles are congruent, then the lines are parallel," we work the other way around:

• $LI$ and $ON$ with transversal $HE$ (given)
• $\angle LAR \cong \angle ARN$ (given)
• $\angle LAR \cong \angle HAI$ (Vertical Angles Theorem)
• $\angle HAI \cong \angle ARN$ (Transitive Property of Congruence)
• $LI \parallel ON$ (Converse of Corresponding Angles Theorem)

Alternate Interior Angles in Real Life

Look at a window with panes divided by muntins. The parallel, vertical muntins are probably transversed by a horizontal muntin. Anywhere they cross, you can find alternate interior angles.

Make a capital letter $Z$, like the Mark of Zorro (you'll probably have to look up that 1919 superhero). The top and bottom horizontal slashes of Zorro's sword are parallel lines, and the diagonal slash is the transversal. Zorro's big $Z$ makes two obvious, alternate interior angles.

Lesson Summary

After navigating this lesson, you are now able to define the parts of an angle (lines, rays or line segments meeting at an endpoint and forming a vertex). You can also draw, describe and identify transversal lines, straight lines, straight angles, parallel lines, and alternate interior angles.

In addition, you can now apply the Alternate Interior Angles Theorem to find angles in parallel lines crossed by a transversal. You definitely figured out the angles on this one!

Next Lesson:

Transversal Liness & Angles