Do you know how to identify corresponding angles?

Jonas found a sculpture made of wire. In the sculpture there were two straight pieces of firm wire that formed parallel lines and one straight piece of wire that intersected the parallel wires.

Jonas counted 7 angles, but he isn't sure this is correct.

Do you know how many angles are formed when a straight line intersects two parallel lines?

**Use this Concept to figure out the solution to Jonas' dilemma.**

### Guidance

We have seen how intersecting lines form four angles that share certain relationships with each other. Now let’s take this idea one step further. 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. We can divide the line which is** \begin{align*}180^\circ\end{align*} **into new angles. Notice the measures of each angle that has been formed.**

**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, we have 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. We call these angles** *corresponding angles.***They are in the same place in each intersection, and they 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. These relationships always exist when a line intersects with parallel lines. Let’s practice identifying corresponding angles.

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

**This time the parallel lines are vertical, but the relationships stay the same. Imagine you could place one intersection on top of each other. They would be exactly the same, and the corresponding angles would be in the same place.**

We need to find the angle that corresponds to angle \begin{align*}Z\end{align*}. 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. Which angle is this?

**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*}.

**Angle** \begin{align*}L\end{align*} **is the top left angle formed at the first intersection. Its corresponding angle will be the top left angle formed at the second intersection. This is angle** \begin{align*}W\end{align*}, **so angles** \begin{align*}L\end{align*} **and** \begin{align*}W\end{align*} **are corresponding angles.**

Now that we understand corresponding relationships, we can use the angles at one intersection to help us find the measure of angles in the other intersection.

As we have said, **corresponding angles are exactly the same, so they have the same measure.**

Therefore **if we know the measure of an angle at one intersection, we also know the measure of its corresponding angle at the second intersection.**

**In the figure above, the** \begin{align*}45^\circ\end{align*} **angle and angle** \begin{align*}A\end{align*} **are corresponding angles. What must the measure of angle** \begin{align*}A\end{align*} **be**? **You guessed it:** \begin{align*}45^\circ\end{align*}. **What about angle** \begin{align*}F\end{align*}**?** **It corresponds to the** \begin{align*}135^\circ\end{align*} **angle in the second intersection, so it too must be** \begin{align*}135^\circ\end{align*}.

*Working in this way is a lot like figuring out a puzzle! You can figure out any missing angles with just a few clues.*

Answer the following questions and figure out the missing angle measures.

#### Example A

If one vertical angle is \begin{align*}45^ \circ\end{align*}, what is the other vertical angle?

**Solution: \begin{align*}45^ \circ\end{align*}**

#### Example B

True or false. Corresponding angles are matching angles.

**Solution: True.**

#### Example C

True or false. When a line intersects two parallel lines, then eight angles are formed.

**Solution: True.**

Here is the original problem once again.

Jonas found a sculpture made of wire. In the sculpture there were two straight pieces of firm wire that formed parallel lines and one straight piece of wire that intersected the parallel wires.

Jonas counted 7 angles, but he isn't sure this is correct.

Do you know how many angles are formed when a straight line intersects two parallel lines?

Jonas isn't correct. When a straight line intersects two parallel lines, then there are 8 angles formed.

### Guided Practice

Here is one for you to try on your own.

Fill in the figure below with the angle measure for all of the angles shown.

**Answer**

Wow, we only have one angle to go on. Not to worry though! We know how to find the measure of its adjacent angles, its vertical angle, and its corresponding angle. That’s all we need to know.

Let’s put in its adjacent angles first. If the known angle is 60, then the adjacent angles are \begin{align*}180 - 60 = 120^\circ\end{align*}. Angle 1 is adjacent to the \begin{align*}60^\circ\end{align*} angle along one line, and angle 3 is adjacent to it along the other line.

Now let’s find the measure of angle 2. It is vertical to the known angle, so we know that these two angles have the same measure. Therefore angle 2 is also \begin{align*}60^\circ\end{align*}. Now we know all of the angles at the first intersection!

Because these lines are parallel, all of the angles at the second intersection correspond to angles at the first intersection. Which angle corresponds to the given \begin{align*}60^\circ\end{align*} angle? Angle 5 does, so it is also \begin{align*}60^\circ\end{align*}. From here, we can either use angle 5 to find the remaining angles (which are either adjacent or vertical to it), or we can use the angles in the first intersection to fill in the corresponding angles. Either way, we can find that angle 4 is \begin{align*}120^\circ\end{align*}, angle 6 is \begin{align*}60^\circ\end{align*}, and angle 7 is \begin{align*}120^\circ\end{align*}. We found all of the angles!

Take a look at your completed drawing. Four angles are \begin{align*}60^\circ\end{align*} and four are \begin{align*}120^\circ\end{align*}. We can change the angle measure to two different numbers, and those numbers will appear exactly the same way.

### Video Review

This Khan Academy video is on angles and parallel lines.

### Explore More

Directions: Define each term.

1. Adjacent Angles

2. Vertical Angles

3. Parallel lines

5. Supplementary angles

6. Complementary angles

7. Corresponding angles.

Directions: Use this diagram to answer the following questions.

8. Are angles D and F vertical angles or corresponding angles?

9. One angle that corresponds to angle D is ?

10. An angle that corresponds to angle E is?

11. True or false. Angle E and angle Q are corresponding angles?

12. True or false. Angle E and angle S are corresponding angles?

13. True or false. Angle E and angle Q are adjacent angles?

14. How many pairs of vertical angles are there in this diagram?

15. Can you find corresponding angles if the intersected lines are not parallel?