Your geometry studies have shown you acute, right and obtuse angles. You may even have learned about straight and reflex angles, but if you are angling to learn even more, you can investigate many other kinds of angles like exterior and interior angles. You can learn about congruent, adjacent, vertical, corresponding, and alternating angles, too.

Before plunging in, let's outline the various angles we can study:

- Congruent angles
- Adjacent angles
- Vertical angles
- Corresponding angles
- Exterior angles
- Consecutive exterior angles
- Alternate exterior angles
- Interior Angles
- Consecutive interior angles
- Alternate interior angles

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.

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 $35\xb0$ angles are congruent, even if they are not identically presented, and are formed with different constructions:

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 $\angle JYO$ adjacent to $\angle OYC$?
- How about $\angle OYC$ adjacent to $\angle KYC$?
- You saw $\angle KYC$ adjacent to $\angle KYJ$, right?
- And you found $\angle KYJ$ adjacent to $\angle JYO$, surely!

In our same drawing above, angles that skip an angle, that is, angles that are not touching each other except at their vertex, are **vertical 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 $\angle JYO$ and $\angle KYC$ made a pair? They touch only at $PointY$
- Did you find $\angle KYJ$ and $\angle OYC$ made the other pair? They also touch only at $PointY$

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.

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 $\angle AYD$ corresponding to $\angle TLY$?
- How about $\angle DYR$ corresponding to &$\angle YLO$?
- You found $\angle RYL$ corresponding to $\angle OLI$, right?
- And you did not overlook $\angle AYL$ corresponding to $\angle TLI$, did you?

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

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:

When the exterior angles are on the same side of the transversal, they are **consecutive exterior angles,** and they are **supplementary** (adding to $180\xb0$). In our figure above, $\angle AYD$ and $\angle TLI$ are consecutive exterior angles. The only other pair of consecutive exterior angles is …

Did you say $\angle DYR$ and $\angle OLI$? We hope so because that's right!

**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 $\angle AYD$ and $\angle OLI$, and then you wrote $\angle DYR$ paired with $\angle TLI$, no doubt!

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

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.

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 $\angle RYL$ pairing off with $\angle YLO$? Did you see that $\angle AYL$ paired up with $\angle TLY$?

In parallel lines, consecutive interior angles are supplementary.

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, $\angle ALY$ is the alternate interior angle for $\angle YLO$, making them congruent. And, of course, $\angle RYL$ pairs off as the alternate interior angle of $\angle TLY$.

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:

*[insert parallel lines MJ and TE and 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 $\angle MCA=4x+3\xb0$ while $\angle EIS=5x-27\xb0$.

You see right away that these two angles, $\angle MCA$ and $\angle EIS$, 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:

$4x+3\xb0=5x-27\xb0$

$3\xb0=x-27\xb0$ (subtract 4x from both sides)

$30\xb0=x$ (add 27° to both sides)

To find our angles, substitute $30\xb0$ for $x$:

$\angle MCA=4x+3\xb0$

$\angle MCA=4(30\xb0)+3\xb0$

$\angle MCA=120\xb0+3\xb0$

$\mathbf{\angle}\mathbf{MCA}\mathbf{=}\mathbf{123}\mathbf{\xb0}$

Though $\angle EIS$ is *supposed* to be congruent, you can still check it:

$\angle EIS=5x-27\xb0$

$\angle EIS=5(30\xb0)-27\xb0$

$\angle EIS=150\xb0-27\xb0$

$\mathbf{\angle}\mathbf{EIS=}\mathbf{123}\mathbf{\xb0}$

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 $\angle JCI$ and $\angle TIS$?

What if we told you $\angle JCI=2y-7\xb0$ while $\angle TIS=y-8\xb0$?

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:

- $\angle JCI$ is the consecutive interior angle partner of $\angle EIC$
- $\angle EIC$ is the vertical angle partner of $\angle TIS$

This means our two problematic angles are actually supplementary, which is a great hint. Together, their two equations must add to $180\xb0$:

$2y-7\xb0+y-8\xb0=180\xb0$ (now simplify)

$3y-15\xb0=180\xb0$ (now add $15\xb0$ to both sides)

$3y=195\xb0$ (now divide by 3)

$\mathbf{y}\mathbf{=}\mathbf{65}\mathbf{\xb0}$

Working back through the problem, you will find that $\angle JCI=123\xb0$ and $\angle TIS=57\xb0$. 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.

Instructor: **Malcolm M.**

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

Get better grades with tutoring from top-rated private tutors. Local and online.

View Tutors

Tutors online

Get better grades with tutoring from top-rated professional tutors. 1-to-1 tailored lessons, flexible scheduling. Get help fast. Want to see the math tutors near you?

Learn faster with a math tutor. Find a tutor locally or online.

Get Started