Converse Alternate Exterior Angles Theorem

Alternate Exterior Angles Definition

Alternate outside angles are created when three lines intersect. A line that crosses two or more other lines is called a transversal. Frequently, ii of the lines will exist parallel, setting upwards some interesting angles with the transversal.

1. Definition
2. Theorem
3. Converse
4. Examples

When a transversal crosses two other lines, it creates an exterior and interior for the parallel lines. Alternate exterior angles are created in the space exterior the parallel lines on alternate sides; interior angles are created in the space within the parallel lines.

Here are lines $TR$ and $IP$, which would definitely cantankerous somewhere in the altitude. We cut across $TR$ and $IP$ with transversal $S\mathrm{Due\; west}$, and where the transversal crosses $TR$ and $IP$, we have Points $\mathrm{50}$ and $O$. Yep, information technology's a $SLOW$ $TRIP$.

We can write each angle with 3 letters: $\angle \mathrm{Southward}LT$, $\angle \mathrm{South}\mathrm{Fifty}R$, and so on down to $\angle POW$.

We can as well employ numbers in these aforementioned vertices, so $\angle SLT$ is $1$, $\angle \mathrm{Due\; south}LR$ is $2$, and so on, ending with $\angle POW$ equally $8$.

The four exterior angles:

1. $\angle 1$
2. $\angle 2$
3. $\angle 7$
4. $\angle \mathrm{viii}$

Alternating exterior angles are pairs that appear outside the crossed lines and on different lines:

• $\angle 1a\mathrm{northward}d\angle 8$
• $\angle 2and\angle 7$

Alternating Exterior Angles Theorem

When two lines are parallel, the transversal creates congruent alternating exterior angles. The theorem says:

The Alternate Exterior Angles Theorem states that if a pair of parallel lines are cut by a transversal, then the alternating exterior angles are coinciding.

Here we have a new pair of lines, parallel and crossed by a transversal. Nosotros are non bothering to identify our parallel lines and transversal! We are jumping ahead to numbered angles.

The exterior angles are these aforementioned four:

1. $\angle 1$
2. $\angle 2$
3. $\angle 7$
4. $\angle 8$

This time, we can apply the Alternate Outside Angles Theorem to state that the alternate exterior angles are congruent:

• $\angle \mathrm{ane}\cong \angle 8$
• $\angle \mathrm{two}\cong \angle 7$

Converse of the Alternating Exterior Angles Theorem

The converse of the Alternate Exterior Angles Theorem is also true:

The Converse of the Alternate Exterior Angles Theorem states that if alternate exterior angles of 2 lines crossed by a transversal are coinciding, then the two lines are parallel.

Alternate Outside Angles Examples

Begin past identifying alternate exterior angles, a common geometry problem. Here is some other set of lines crossed by a transversal, with numbered angles:

Which are exterior angles?

Which are alternate exterior angles?

To detect exterior angles, look in the space in a higher place and beneath the crossed lines. To find alternate exterior angles, look at that outside infinite for each crossed line, on different sides of the transversal.

We hope yous said $\angle 1$, $\angle 2$, $\angle \mathrm{seven}$, and $\angle \mathrm{eight}$ are the exterior angles. We also hope you lot identified $\angle 1$ and $\angle 8$ as a pair of alternating outside angles and $\angle 2$ with $\angle 7$ as the other pair.

Another type of problem is to identify the other angle that pairs with a given alternating outside bending. Hither are more than lines and a transversal, with numbered angles:

$\angle \mathrm{ii}$ is an exterior angle; what is its alternate exterior bending?

To find the partner of $\angle 2$, look on the other side of the transversal. Expect outside the crossed lines. Wait at a crossed line that does not contain $\angle \mathrm{two}$.

Did you say $\angle 7$? We hope so!

Example With Parallel Lines

Here is a drawing with parallel lines and a transversal.

If nosotros know that $\angle \mathrm{eight}$ measures $130°$, what is the mensurate of $\angle 1$?

$\angle \mathrm{viii}$ is on the outside of the bottom parallel line, and to the correct of the transversal. That means $\angle 1$ is its alternate exterior bending partner. The Alternating Exterior Angles Theorem tells the states it is as well $130°$!

Lesson Summary

Now that you lot have gone through this lesson carefully, yous are able to recall that angles on opposite sides of a transversal and outside two lines are chosen alternating exterior angles. When lines crossed past the transversal are parallel, you can apply the Alternate Exterior Angles Theorem to know the alternating exterior angles are congruent. You can now solve issues identifying and measuring alternate exterior angles.

Geometry is a bit similar a sorcerer'due south trick. It sets up some odd upshot, like parallel lines being crossed by a transversal, and so hopes you're astounded when that improbable outcome leads to something new, like alternate outside angles. Even when the lines are not parallel, alternate exterior angles exist, and nosotros don't need to pull a rabbit out of a lid to amaze you with them.

Next Lesson:

Alternate Interior Angles

Converse Alternate Exterior Angles Theorem,

Source: https://tutors.com/math-tutors/geometry-help/alternate-exterior-angles-definition-theorem

Posted by: baylesthabelloved.blogspot.com

Share this post