What Are Adjacent Angles?

⭐ An angle is made up of two rays that share a common vertex.

$sample angle$

The angle ∠XYZ above has two arms: ray YX and ray YZ.

Its vertex is point Y.

⭐ Adjacent angles are two angles that share a common vertex and a common arm. They share no interior points.

$adjacent angles$

The word "adjacent" means side-by-side, or directly next to.

This means that adjacent angles are side-by-side angles.

Sharing a Vertex

Look at the two angles below.

$angle ABC$

The first angle is ∠ABC.

What is its vertex?

Yes, it's point B.

The second angle is ∠CBD.

$angle CBD$

What is its vertex?

Correct. It's point B as well.

The two angles have the same vertex, point B.

Sharing a Common Side

Adjacent angles also share a common side.

The two angles below share a common side, ray BC.

$adjacent angles$

Sharing No Interior Points

Adjacent angles do not share any interior points.

All the points within ∠ABC are on one side.

$interior points for angle ABC$

The points within ∠CBD are on the other side.

$interior points of angle CBD$

None of the points overlap.

This means these two angles are adjacent. ✅

Angles That Are Not Adjacent

$angles DEF and GHI$

Do these angles have a common vertex?

No, they don't.

Do they have a common side?

No, they don't.

They're not adjacent.

Adding Adjacent Angles

Let's look at these adjacent angles.

$adjacent angles$

When we put ∠ABC and ∠CBD together, they create the bigger angle ∠ABD.

Since the two smaller angles are adjacent, we can add up their measurements to get the measurement of the bigger angle.

∠ABC + ∠CBD = ∠ABD

Let's take a look at the measurements of the two angles:

$adjacent angles$

What is the size of the new, bigger angle ∠ABD?

140° + 40° = 180°

Yes it's 180°. ✅

$adjacent angles$

Tip:

⭐ A straight angle measures exactly 180°.

Subtracting to Find Adjacent Angles

What is the value of n below?

$adjacent angles$

We know the sum of the angles is 75°.

One angle is 25°.

How do we find n?

We subtract the size of the angle we know from the total.

n = total - known angle

Let's subtract:

n = 75° - 25°

n = 50°

The size of the unknown angle is 50°. ✅

Let's check by adding the angles:

50° + 25° = 75°

Fantastic. Our answer is correct.

$adjacent angles$

Great job learning about adjacent angles.

Let's Review

⭐ Adjacent angles share a common vertex and side, but no interior points.

Do you think you can spot adjacent angles on your own?

Try the practice to see.

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