Malik is following an old map of the downtown area. He decides to draw his own map and label his favorite places so that he doesn't forget where to stop when he visits the city again. He notices that the original map looks like a lot of parallel and perpendicular lines with other lines as well. He wonders if he can figure out the angle for areas that are in similar positions in reference to the lines. He notices that his current corner looks similar to the next corner. In fact, the two streets appear to be parallel with a line connecting them. If the angle of one corner is \begin{align*}78^o\end{align*}, what should Malik expect to be the measure of the next corner?

In this concept, you will learn how to identify and use corresponding angles.

### Identifying and Using Corresponding Angles

When a line intersects with two lines that are parallel, it forms the same angles of intersection with the first parallel line and the second. Let’s see what this looks like.

Notice that new angle relationships are formed and take note of the measures of each angle.

When line \begin{align*}y\end{align*} intersects with line \begin{align*}a\end{align*}, it forms \begin{align*}100^\circ\end{align*} angles and \begin{align*}80^\circ\end{align*} angles. When it intersects with line \begin{align*}b\end{align*}, it also forms \begin{align*}100^\circ\end{align*} and \begin{align*}80^\circ\end{align*} angles. This is because lines \begin{align*}a\end{align*} and \begin{align*}b\end{align*} are parallel. Any line will intersect with them the same way.

In this situation, there is another angle relationship that will help us find the measure of the angles formed at either point of intersection. Every angle at the first intersection (between lines \begin{align*}y\end{align*} and \begin{align*}a\end{align*}) corresponds to an angle at the second intersection (between lines \begin{align*}y\end{align*} and \begin{align*}b\end{align*}). It occurs in the same place and has the same measure. Take a look at the figure below.

Angle \begin{align*}E\end{align*} in the first intersection is in the same place as angle \begin{align*}Q\end{align*} in the second intersection. These angles are **corresponding angles .** Corresponding angles are in the same place in each intersection and have the same measure. Angles \begin{align*}D\end{align*} and \begin{align*}P\end{align*} are corresponding angles, angles \begin{align*}G\end{align*} and \begin{align*}S\end{align*} are corresponding, and angles \begin{align*}F\end{align*} and \begin{align*}R\end{align*} are corresponding.

Let’s look at an example.

What angle corresponds to angle \begin{align*}Z\end{align*}?

This time the parallel lines are vertical, but the relationships stay the same.

Angle \begin{align*}Z\end{align*} is the bottom right angle formed at the second intersection. Its corresponding angle will be the bottom right angle formed at the first intersection. Angle \begin{align*}O\end{align*} occurs at the same place in the first intersection, so it is the corresponding angle to angle \begin{align*}Z\end{align*}.

### Examples

#### Example 1

Earlier, you were given a problem about Malik and his map.

Malik wants to recreate the map so that only his favorite places are included. He is standing at the corner of a street that runs parallel to the next street and is intersected by the third street. If the angle of his current corner is \begin{align*}78^o\end{align*}, what will be the angle measure of the next corner when Malik draws it on his new map?

First, identify the location of the corners.

The corner is at an intersection of two streets and the next corner is at a parallel intersection.

Next, name the relationship between the angles of the two corners.

The corners have corresponding angles.

Then, name the measure of the next corner's angle.

\begin{align*}78^o\end{align*}

The answer is that the angle measure of the next corner will be \begin{align*}78^o\end{align*}.

#### Example 2

Solve the following problem.

In the figure above, the \begin{align*}45^\circ\end{align*} angle and angle \begin{align*}A\end{align*} are corresponding angles. What must be the measure of angle \begin{align*}A\end{align*}?

First, identify the angle that corresponds to angle \begin{align*}A\end{align*}.

The \begin{align*}45^o\end{align*} angle corresponds to angle \begin{align*}A\end{align*}.

Next, remember the relationship between corresponding angles.

Corresponding angles are equal.

Then, state the value of angle \begin{align*}A\end{align*}.

Angle \begin{align*}A\end{align*} is \begin{align*}45^o\end{align*}.

The answer is that the measure of angle \begin{align*}A\end{align*} is \begin{align*}45^o\end{align*}.

#### Example 3

What is the measure of angle \begin{align*}F\end{align*}?

First, identify the angle that corresponds to angle \begin{align*}F\end{align*}.

The \begin{align*}135^\circ\end{align*} angle corresponds to angle \begin{align*}F\end{align*}.

Next, remember the relationship between corresponding angles.

Corresponding angles are equal.

Then, state the value of angle \begin{align*}F\end{align*}.

Angle \begin{align*}F\end{align*} is \begin{align*}135^\circ\end{align*}.

The answer is that the measure of angle \begin{align*}F = 135^o\end{align*}.

#### Example 4

What angle corresponds to angle \begin{align*}L\end{align*}?

First, identify the location of the angle.

Angle \begin{align*}L\end{align*} is the top left angle formed at the first intersection.

Next, identify the location of the corresponding angle.

The corresponding angle will be the top left angle formed at the second intersection.

Then, name the corresponding angle.

The corresponding angle is angle \begin{align*}W\end{align*}.

The answer is that angle \begin{align*}W\end{align*} corresponds to angle \begin{align*}L\end{align*}.

#### Example 5

What is the value of angle 5?

First, identify the angle that corresponds to angle 5.

Angle 5 corresponds with the given\begin{align*}60^o\end{align*} angle.

Next, remember the relationship between corresponding angles.

Corresponding angles are equal.

Then, state the value of angle 5.

Angle 5 is \begin{align*}60^o\end{align*}.

The answer is that the measure of angle 5 is \begin{align*}60^o\end{align*}.

### Review

Define each term.

- Adjacent Angles
- Vertical Angles
- Parallel lines
- Perpendicular lines
- Supplementary angles
- Complementary angles
- Corresponding angles.

Use this diagram to answer the following questions.

- Are angles D and F vertical angles or corresponding angles?
- One angle that corresponds to angle D is ?
- An angle that corresponds to angle E is?
- True or false. Angle E and angle Q are corresponding angles?
- True or false. Angle E and angle S are corresponding angles?
- True or false. Angle E and angle Q are adjacent angles?
- How many pairs of vertical angles are there in this diagram?
- Can you find corresponding angles if the intersected lines are not parallel?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 8.6.