8.6: Corresponding Angles
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
When line
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
Angle
What angle corresponds to angle
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
Angle
Angle
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
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
Solution:
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.
Vocabulary
Here are the vocabulary words in this Concept.
 Adjacent Angles

angles formed by intersecting lines that are supplementary and are next to each other. The sum of their angles is
180∘ .
 Vertical Angles
 angles formed by intersecting lines that are on the diagonals. They have the same measure.
 Supplementary

having a sum of
180∘ .
 Intersecting Lines
 lines that cross at one point.
 Parallel Lines
 lines that will never cross.
 Perpendicular Lines
 lines that intersect at a right angle.
 Corresponding Angles
 Angles that are in the same place in each intersection when a line crosses two parallel lines.
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
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
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
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
Here is a video for review.
 This Khan Academy video is on angles and parallel lines.
Practice
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 are 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?
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Image Attributions
Here you'll learn to identify corresponding angles formed when a line intersects two other lines.