Written by

Malcolm McKinsey

Fact-checked by

Paul Mazzola

## Types of angles

In geometry, there are many types of angles such as congruent, adjacent, vertical, corresponding, alternating, exterior, and interior angles. All angles have relationships to other angles and those angle relationships are what we will cover here.

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## Angle relationships

Beyond measuring the degrees or radians, you can also compare angles and consider their relationships to other angles. We talk of angle relationships because we are comparing position, measurement, and congruence between two or more angles.

For example, when two lines or line segments intersect, they form two pairs of vertical angles. When two parallel lines are intersected by a transversal, complex angle relationships form, such as alternating interior angles, corresponding angles, and so on.

Being able to spot angle relationships, and confidently find congruent angles when lines intersect, will make you a better, geometry student. You will solve complex problems faster when you are thoroughly familiar with all the types of angle relationships.

### Congruent angles

Any two angles, no matter their orientation, that have equal measures (in radians or degrees) are**congruent**. They show the same "openness" between the two rays, line segments or lines that form them. So these two**35°**angles are congruent, even if they are not identically presented, and are formed with different constructions:

### Adjacent angles

When two lines cross each other, they form four angles. Any two angles sharing a ray, line segment or line are**adjacent**. In the following drawing,**LineJC**intersects**LineOK**, creating four adjacent pairs and intersecting at**PointY**. Can you find them all?

Did you find

**∠JYO**adjacent to**∠OYC**?How about

**∠OYC**adjacent to**∠KYC**?You saw

**∠KYC**adjacent to**∠KYJ**, right?And you found

**∠KYJ**adjacent to**∠JYO**, surely!

### Vertical angles

In our same drawing above, angles that skip an angle, that is, angles that are not touching each other except at their vertex, arevertical angles. Here the word "vertical" means "relating to a vertex," not "up and down." Vertical angles are opposite angles; they share only their vertex point.

Two intersecting lines create two pairs of vertical angles. See if you can spot them in our drawing.

Did you find

**∠JYO**and**∠KYC**made a pair? They touch only at**Point Y**Did you find

**∠KYJ**and**∠OYC**made the other pair? They also touch only at**Point Y**

You may wonder why adjacent angles are not also vertical angles, since they share the vertex, too. Adjacent angles share more than the vertex; they share a common side to an angle.

### Corresponding angles

Anytime a transversal crosses two other lines, we get corresponding angles. The more restrictive our intersecting lines get, the more restrictive are their angle relationships. When a line crosses two parallel lines (a transversal), a whole new level of angle relationships opens up:

We can** adroitly**pull from this figure angles that look like each other. Angles that have the same position relative to one another in the two sets of four angles (four at the top,

**LineAR**; four at the bottom,

**LineTO**) are corresponding angles. When the corresponding angles are on parallel lines, they are congruent.

Our transversal and parallel lines create four pairs of corresponding angles. Can you name them all?

Did you find

**∠AYD**corresponding to**∠TLY**?How about

**∠DYR**corresponding to**∠YLO**?You found

**∠RYL**corresponding to**∠OLI**, right?And you did not overlook

**∠AYL**corresponding to**∠TLI**, did you?

In all cases, since our**LineAR**and**TO**are parallel, their corresponding angles are congruent.

### Exterior angles

Those same parallel lines and their transversal create exterior angles. An exterior angle among line constructions (not polygons) is one that lies outside the parallel lines. You can see two types of exterior angle relationships:

### Consecutive exterior angles

When the exterior angles are on the same side of the transversal, they are**consecutive exterior angles,**and they are**supplementary**(adding to**180°**). In our figure above,**∠AYD**and**∠TLI**are consecutive exterior angles. The only other pair of consecutive exterior angles is **∠DYR**and**∠OLI.**

### Alternate exterior angles

**Alternate exterior angles**are similar to vertex angles, in that they are opposite angles (on either side of the transversal). Alternate exterior angles are on opposite sides of the transversal (that's the alternate part) and outside the parallel lines (that's the exterior part). Can you find the two pairs of alternate exterior angles in our drawing?

You wrote down**∠AYD**and**∠OLI**, and then you wrote**∠DYR**paired with**∠TLI**, no doubt!

Congruent alternate exterior angles are used to prove that lines are parallel, using (fittingly) the Alternate Exterior Angles Theorem.

### Interior angles

Angles between the bounds of the two parallel lines are**interior angles**, again created by the transversal. Just as with exterior angles, we can have consecutive interior angles and alternate interior angles.

### Consecutive Interior Angles

Interior angles on the same side of the transversal are**consecutive interior angles**. In our figure, can you find the two pairs? Did you find**∠RYL**pairing off with**∠YLO**? Did you see that**∠AYL**paired up with**∠TLY**?

In parallel lines, consecutive interior angles are supplementary.

### Alternate interior angles

When the interior angles are on opposite sides of the transversal, they are**alternate interior angles**. They lend themselves to the Alternate Interior Angles Theorem, which states that congruent alternate interior angles prove parallel lines (much as the Alternate Exterior Angles Theorem did).

In our figure,**∠ALY**is the alternate interior angle for**∠YLO**, making them congruent. And, of course,**∠RYL**pairs off as the alternate interior angle of**∠TLY**.

## Angle relationships examples

You can use your newfound knowledge of angle relationships to solve algebraic challenges about geometric figures. When viewing any new figure, go through your list and determine three things:

Relative positions of the two questioned angles

Whether the angles are outside the parallel lines (exterior) or inside the parallel lines (interior)

Whether the two angles under investigation are on the same side of the transversal (consecutive) or opposite sides of the transversal (alternate)

Once you understand the relationship between the two angles, you can assume some basic facts, such as their congruence or that they may be supplementary.

You can use that awareness to solve seemingly difficult algebraic problems like this:

Draw **parallel** **lines MJ** and **TE **and the **transversal AS** with intersecting **Point C** on **Line MJ **and intersecting **Point I **on **Line TE**, spelling in a circular way **MAJESTIC. **Let **∠MCA = 123°**.

Given the figure, find the value of** x**if∠

**MCA=4x+3°**while

**∠EIS=5x−27°**.

You see right away that these two angles,**∠MCA** and**∠EIS**, are exterior angles on opposite sides of the transversal. So they are alternate exterior angles, making them congruent and allowing you to set up a simple algebraic equation:

$4x+3°=5x−27°$4x+3°=5x−27°

Next, we subtract **4*** x* from both sides:

$3°=x−27°$3°=x−27°

Next, we add **27°** to both sides:

$30°=x$30°=x

To find our angles, substitute**30°**for* x*:

$∠MCA=4x+3°$∠MCA=4x+3°

$∠MCA=4(30°)+3°$∠MCA=4(30°)+3°

$∠MCA=120°+3°$∠MCA=120°+3°

$∠MCA=123°$∠MCA=123°

Though**∠EIS**is*supposed*to be congruent, you can still check it:

$∠EIS=5x−27°$∠EIS=5x−27°

$∠EIS=5(30°)−27°$∠EIS=5(30°)−27°

$∠EIS=150°−27°$∠EIS=150°−27°

$∠EIS=123°$∠EIS=123°

Let's try a second exercise, using the same figure. If you can solve this, you will have accomplished some**MAJESTIC**mathematics!

What can you tell us about**∠JCI**and**∠TIS**?

What if we told you**∠JCI=2y−7°**while**∠TIS=y−8°**?

Can you solve for**y**?

Neither angle is on the same side of the transversal, nor are they both outside the parallel lines. They are not both inside the parallel lines, either! They seem to have no relationship at all!

And yet, by deduction, you can see a relationship:

**∠JCI**is the consecutive interior angle partner of**∠EIC****∠EIC**is the vertical angle partner of**∠TIS**

This means our two problematic angles are actually supplementary, which is a great hint. Together, their two equations must add to**180°**:

$2y−7°+y−8°=180°$2y−7°+y−8°=180°

Now, simplify:

$3y−15°=180°$3y−15°=180°

Need**15°**to both sides:

$y=65°$y=65°

Working back through the problem, you will find that**∠JCI=123°**and**∠TIS=57°**. Success! Practice for yourself, by constructing parallel lines with transversals and identifying all the angle relationships they create.

Remember, too, the relationships still hold when the lines cut by the transversal are not parallel; you just cannot use Theorems to make assumptions about the angles.