Alternate Exterior Angles Converse Proof

Alternating Exterior Angles Definition

Alternate exterior angles are created when three lines intersect. A line that crosses two or more than other lines is called a transversal. Oftentimes, two of the lines will be parallel, setting up some interesting angles with the transversal.

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

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

Hither are lines $TR$ and $IP$, which would definitely cantankerous somewhere in the distance. We cut across $TR$ and $IP$ with transversal $SW$, and where the transversal crosses $TR$ and $IP$, we have Points $L$ and $O$. Aye, information technology'due south a $SLO\mathrm{Westward}$ $TRIP$.

We can write each angle with 3 letters: $\angle SLT$, $\angle \mathrm{Southward}LR$, and and then on down to $\angle POW$.

We tin also use numbers in these same vertices, so $\angle \mathrm{Due\; south}LT$ is $1$, $\angle S\mathrm{Fifty}R$ is $2$, then on, catastrophe with $\angle PO\mathrm{Westward}$ as $8$.

The four exterior angles:

1. $\angle 1$
2. $\angle 2$
3. $\angle \mathrm{seven}$
4. $\angle 8$

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

• $\angle 1and\angle 8$
• $\angle \mathrm{ii}a\mathrm{due\; north}d\angle \mathrm{vii}$

Alternate Exterior Angles Theorem

When 2 lines are parallel, the transversal creates congruent alternate exterior angles. The theorem says:

The Alternate Outside Angles Theorem states that if a pair of parallel lines are cutting by a transversal, then the alternate exterior angles are congruent.

Here we have a new pair of lines, parallel and crossed by a transversal. Nosotros are not bothering to identify our parallel lines and transversal! We are jumping ahead to numbered angles.

The exterior angles are these same four:

1. $\angle \mathrm{ane}$
2. $\angle \mathrm{two}$
3. $\angle 7$
4. $\angle 8$

This fourth dimension, we can use the Alternate Outside Angles Theorem to state that the alternate outside angles are congruent:

• $\angle 1\cong \angle 8$
• $\angle 2\cong \angle 7$

Converse of the Alternate Exterior Angles Theorem

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

The Converse of the Alternate Outside Angles Theorem states that if alternating exterior angles of two lines crossed by a transversal are congruent, then the two lines are parallel.

Alternate Outside Angles Examples

Begin by identifying alternate outside angles, a common geometry trouble. Hither is another set up of lines crossed by a transversal, with numbered angles:

Which are exterior angles?

Which are alternate exterior angles?

To find exterior angles, look in the space above and beneath the crossed lines. To find alternating exterior angles, await at that exterior space for each crossed line, on different sides of the transversal.

We hope yous said $\angle 1$, $\angle \mathrm{ii}$, $\angle 7$, and $\angle 8$ are the outside angles. We also hope y'all identified $\angle 1$ and $\angle 8$ as a pair of alternating exterior angles and $\angle \mathrm{two}$ with $\angle 7$ as the other pair.

Another type of problem is to identify the other angle that pairs with a given alternate exterior bending. Here are more lines and a transversal, with numbered angles:

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

To notice the partner of $\angle 2$, wait on the other side of the transversal. Look outside the crossed lines. Look at a crossed line that does not incorporate $\angle \mathrm{ii}$.

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

Example With Parallel Lines

Hither is a drawing with parallel lines and a transversal.

If we know that $\angle 8$ measures $130°$, what is the measure out of $\angle \mathrm{one}$?

$\angle \mathrm{eight}$ is on the outside of the lesser parallel line, and to the right of the transversal. That means $\angle \mathrm{ane}$ is its alternate exterior angle partner. The Alternate Outside Angles Theorem tells us it is likewise $130°$!

Lesson Summary

Now that you take gone through this lesson carefully, you are able to recall that angles on contrary sides of a transversal and outside two lines are called alternate exterior angles. When lines crossed by the transversal are parallel, you can utilize the Alternate Exterior Angles Theorem to know the alternate exterior angles are coinciding. You tin now solve problems identifying and measuring alternating outside angles.

Geometry is a fleck like a sorcerer's play a joke on. It sets up some odd consequence, similar parallel lines being crossed by a transversal, and then hopes yous're astounded when that improbable event leads to something new, similar alternating outside angles. Even when the lines are not parallel, alternate outside angles exist, and nosotros don't need to pull a rabbit out of a hat to amaze y'all with them.

Next Lesson:

Alternate Interior Angles

Alternate Exterior Angles Converse Proof,

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

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