Want more practice solving with angle pairs? How about more review for solving angles in triangles? Look no further. Get more practice here, and test your ability with a quiz.

## Solving Angles

In another lesson, we learned about the different types of angles: consecutive interior, alternate interior, alternate exterior and corresponding. We discovered that when two lines are parallel, all of the angle pairs are congruent, except for consecutive interior angles, which are supplementary. We also learned about vertical angles, which are always congruent.

Let’s do some practice with these angles.In the figure below, let *Angle 5* = 30*y* + 31, and let *Angle 9* = 22*y* + 55. What is the value of *y*?

*Angle 5* and *Angle 9* don’t match any of the angle pairs, so let’s find the connection between their measures. We notice that *Angle 5* corresponds to *Angle 1*, and *Angle 1* corresponds to *Angle 9*.

Knowing that all corresponding angles are congruent, *Angle 5* = *Angle 1*, and *Angle 1* = *Angle 9*. So, by the transitive property of equality, we can conclude that *Angle 5* = *Angle 9*. By substituting the equations, we have 30*y* + 31 = 22*y* + 55. From here, we can subtract 31 from both sides to get 30*y* = 22*y* + 24, and then subtract 22*y* from each side to get 8*y* = 24. To finish, we will divide both sides by 8 to determine that *y* = 3.Let’s do another using the same figure.

This time, let *Angle 4* = 14*x* – 23, and let *Angle 14* = 4*x* + 5. Find the measure of *Angle 15*.Once again, these angles are not a special angle pair; so, let’s find the connection. *Angle 4* corresponds to *Angle 12*, and *Angle 12* is consecutive interior to *Angle 14*. Therefore, *Angle 4* = *Angle 12*, and *Angle 12* + *Angle 14* = 180. With this knowledge, we can replace *Angle 12* with *Angle 4* to get *Angle 4* + *Angle 14* = 180.

With the equations, we have 14*x* – 23 + 4*x* + 5 = 180. Combining like terms gives us 18*x* – 18 = 180, and then, by adding 18 to both sides, we get 18*x* = 198. Last, we will divide both sides by 18 to conclude that *x* = 11.

Now we can find the value of *Angle 15*, which is vertical to and congruent with *Angle 14*. Substituting 11 into the equation, we see that *Angle 14* = 4(11) + 5, which equals 49. Therefore, we can also conclude that *Angle 15* = 49 degrees.

## Solving Triangles

When working with triangles, remember that the sum of all three angles in every triangle is 180 degrees. Let’s get started.In this first triangle below, let’s solve for *x*.

For each angle, we either have a measure or an equation. For that reason, let’s add all of the angles together to equal 180 degrees. Doing so, we have 40 + 10*x* + 20 + 20 = 180, and by combining like terms, we have 10*x* + 80 = 180. Next, let’s subtract 80 from both sides to get 10*x* = 100, and then let’s divide each side by 10 to finish with *x* = 10.

Here’s another. This is triangle *JKL*. What is the measure of *Angle L*?

By having information for all three angles, we will add them together to equal 180. Remember that the square in the angle tells us that the angle measures 90 degrees. So, we can begin with 10*y* + 5 + 90 + 15*y* + 35 = 180. Combining like terms gives us 25*y* + 130 = 180.

From here, we will subtract 130 from both sides, leaving 25*y* = 50. Then, let’s divide both sides by 25 to see that *y* = 2. Now, by substitution, we see that *Angle L* = 15(2) + 35, which equals 65 degrees.Let’s do one more.

In triangle *DEF* below, *Angle D* is two times a number, *Angle E* is forty more than five times the number, and *Angle F* is five more than two times the number. What is the measure of *Angle E*?

Since we don’t have the exact equation for each angle, we have to use the descriptions to create them. All of the descriptions reference some unknown number. Not knowing what this number is, we will call it *x* in each equation.

Therefore, *Angle D* = 2*x*, *Angle E* = 5*x* + 40, and *Angle F* = 2*x* + 5.To solve, we begin with 2*x* + 5*x* + 40 + 2*x* + 5 = 180. Combining like terms gives us 9*x* + 45 = 180, and then subtracting 45 from both sides leaves us with 9*x* = 135. From here, we will divide both sides by 9 to get *x* = 15. To complete the problem, we will substitute 15 into the equation for *Angle E* to see that *Angle E* = 5(15) + 40, which equals 115 degrees.

## Lesson Summary

In review, when solving with angles and lines, always begin by determining the special angle pair or the connection between the angles you were given.

This will send you in the right direction for determining whether you should set the angles equal to each other or add them together to equal 180 degrees. But remember, to solve these problems in this manner, the two lines must be parallel.When it comes to triangles, remember that all of the angles in every triangle must add together to equal 180 degrees. So, when you have information for each angle in a triangle, this is the best way to start and solve the problem.

## Learning Outcomes

After completing this lesson, you’ll be able to:

- Solve for angles using the rules for consecutive interior, alternate interior, alternate exterior and corresponding angles
- Calculate an angle of a triangle when given some information about all three angles