A transversal is a line that crosses two or more lines. When the lines crossed by a transversal are parallel, we can figure out the values of all the angles formed with just one given angle.

To do this, we like to imagine the set of lines as a chess board so that the colors can help us identify which angles are equal.

$undefined_{∘}$

$x_{∘}$

angles equal to the given angle

angles equal to the supplementary angle of the given $(180_{∘}−given angle)$

In the same way that alternating squares on a chess board are the same color, alternating angles are equal.

Check out our step-by-step calculator or continue the lesson to explore this in more detail!

Transversals, Parallel Lines & Angles Calculator

Which set of lines best matches the ones in your question?

Step 1. Imagine the diagram as a chess board & identify a given angle.

Click or tap on an angle that you already know. You might know more than one, but we only need one to get started.

$undefined_{∘}$

$x_{∘}$

$angle measure:$

$_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Equal angles make a “zig zag” along the same square color. So, all the purple angles are $0_{∘}$.

If you've found the unknown angle, stop 🛑. If not, keep going.

$null_{∘}$

$x_{∘}$

Step 3. Find the supplementary angle of the given angle.

Since we still haven't found the unknown, we can use supplementary angles to find the angle right next to our given.

$supplementary angle =180_{∘}−0_{∘}=180_{∘} $So, the green angle is $180_{∘}$.

$null_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Step 4. Identify all the angles that are equal to the supplementary angle.

Equal angles make a “zig zag” along the same square color. This gets us the rest of the angle values.

$null_{∘}$

$x_{∘}$

So, if the unknown angle is purple, it is $0_{∘}$. If the unknown angle is green, it is $180_{∘}$.

Key Steps 🗝 How to find Angles between a Transversal & Parallel Lines

Step 1. Imagine the diagram as a chess board & identify a given angle.

You might be given more than one, but only one is needed to get started.

$undefined_{∘}$

$x_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Equal angles make a “zig zag” along the same square color.

$undefined_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Step 3. Find the supplementary angle of the given angle.

$supplementary angle=180_{∘}−given angle$

$undefined_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Step 4. Identify all the angles that are equal to the supplementary angle.

Equal angles make a “zig zag” along the same square color.

$undefined_{∘}$

$x_{∘}$

Continue to explore how to identify angles through some examples and practice problems!

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Finding Angles between a Transversal & Parallel Lines

When a transversal crosses parallel lines, we can imagine the intersection as a chess board and use the colors to figure out all the angles using just one given angle.

$50_{∘}$

$x_{∘}$

$50_{∘}$

$x_{∘}$

We start by figuring out which angles are equal to the given angle. To do this, we just need to see which ones fall on the same chess board color as the given angle.

We got you! To figure out the rest of the angles, we first need to find the value of the angle directly next to the given angle.

$50_{∘}$

$x_{∘}$

This angle is supplementary to the given angle because, together, the two form a straight line equal to 180°. We can use the following equation to find the value of the missing supplementary angle:$supplementary angle=180_{∘}−given angle$Once we have the value of the supplementary angle, we know that every other angle that falls on the same chess board color as the supplementary angle has the same value.Checkmate! We found all the angles!

It’s important to remember that this only works when the transversal crosses parallel lines. If the lines aren’t parallel, we have to apply our knowledge of supplementary and vertical angles to find the missing angles.

Vertical Angles

Vertical angles are the angles opposite each other when two lines intersect. Vertical angles are special because no matter what, vertical angle pairs are always equal.

$∠a=∠c∠b=∠d∠e=∠g∠f=∠h$

Nice work! Now let's try some example problems together.

Let's start by finding the unknown angle $x$ below ⬇️.

$40_{∘}$

$x_{∘}$

Step 1. Imagine the diagram as a chess board & identify a given angle.

We can imagine our angles in an alternating pattern to help us out.

$40_{∘}$

$x_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Equal angles make a “zig zag” along the same square color. So, all the purple angles are $40_{∘}$.

If you've found the unknown angle, stop 🛑. If not, keep going.

$40_{∘}$

$x_{∘}$

Since the unknown angle is one of the purple angles, we know the unknown is $x=40_{∘}$.

Step 3. Find the supplementary angle of the given angle.

Since we still haven't found the unknown, we can use supplementary angles to find the angle right next to our given.

$supplementary angle =180_{∘}−40_{∘}=140_{∘} $So, the green angle is $140_{∘}$.

$40_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Step 4. Identify all the angles that are equal to the supplementary angle.

Equal angles make a “zig zag” along the same square color. This gets us the rest of the angle values.

$40_{∘}$

$x_{∘}$

So, if the unknown angle is purple, it is $40_{∘}$. If the unknown angle is green, it is $140_{∘}$.

Since the unknown angle is green, we know the unknown is $x=140_{∘}$.

Nice work! Now try finding the unknown angle $x$ below ⬇️.

$50_{∘}$

$x_{∘}$

Step 1. Imagine the diagram as a chess board & identify a given angle.

We can imagine our angles in an alternating pattern to help us out.

$50_{∘}$

$x_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Equal angles make a “zig zag” along the same square color. So, all the purple angles are $50_{∘}$.

If you've found the unknown angle, stop 🛑. If not, keep going.

$50_{∘}$

$x_{∘}$

Step 3. Find the supplementary angle of the given angle.

Since we still haven't found the unknown, we can use supplementary angles to find the angle right next to our given.

$supplementary angle =180_{∘}−50_{∘}=130_{∘} $So, the green angle is $130_{∘}$.

$50_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Since the unknown angle is the green supplementary angle, we know the unknown is $x=130_{∘}$.

Step 4. Identify all the angles that are equal to the supplementary angle.

Equal angles make a “zig zag” along the same square color. This gets us the rest of the angle values.

$50_{∘}$

$x_{∘}$

So, if the unknown angle is purple, it is $50_{∘}$. If the unknown angle is green, it is $130_{∘}$.

Since the unknown angle is green, we know the unknown is $x=130_{∘}$.

Amazing! Let's try a harder one now. Find the unknown angle $x$ below ⬇️.

$110_{∘}$

$x_{∘}$

Step 1. Imagine the diagram as a chess board & identify a given angle.

We can imagine our angles in an alternating pattern to help us out.

$110_{∘}$

$x_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Equal angles make a “zig zag” along the same square color. So, all the purple angles are $110_{∘}$.

If you've found the unknown angle, stop 🛑. If not, keep going.

$110_{∘}$

$x_{∘}$

Step 3. Find the supplementary angle of the given angle.

Since we still haven't found the unknown, we can use supplementary angles to find the angle right next to our given.

$supplementary angle =180_{∘}−110_{∘}=70_{∘} $So, the green angle is $70_{∘}$.

$110_{∘}$

$x_{∘}$

If you've found the unknown angle, stop 🛑. If not, keep going.

Step 4. Identify all the angles that are equal to the supplementary angle.

Equal angles make a “zig zag” along the same square color. This gets us the rest of the angle values.

$110_{∘}$

$x_{∘}$

So, if the unknown angle is purple, it is $110_{∘}$. If the unknown angle is green, it is $70_{∘}$.

Since the unknown angle is green, we know the unknown is $x=70_{∘}$.

If you’d like additional practice with finding angles between a transversal and parallel lines, take a moment to complete the quick practice below before closing out this lesson! ⚡

Practice: Finding Angles Between a Transversal & Parallel Lines

Question 1 of 8: Find the unknown angle $x$ below ⬇️:

$40_{∘}$

$x_{∘}$

Step 1. Imagine the diagram as a chess board & identify the given angle.

Tap/click on the angle we know:

$undefined_{∘}$

$x_{∘}$

What is the measure of the given angle?$_{∘}$

Step 2. Identify all the angles that are equal to the given angle.

Remember, equal angles make a “zig zag” along the same square color.

$undefined_{∘}$

$x_{∘}$

That's right! All the purple angles are equal to our given angle, which means our unknown angle is$_{∘}$.

Step 4. Identify all the angles that are equal to the supplementary angle.

Remember, equal angles make a “zig zag” along the same square color.

$undefined_{∘}$

$x_{∘}$

That's right! So, if the unknown angle is purple, it is $40_{∘}$. If the unknown angle is green, it is $140_{∘}$.

Given this, what is the value of our unknown angle?$_{∘}$

Great work! The unknown angle is $40_{∘}$.

When you feel like you've mastered this lesson, click for a celebration ⬇️!

Incredible job, look at you go! Thanks for checking out this lesson ☺️🙏. Where to next?