8.2: Special Pairs of Angles
Introduction
At the Museum
When the students arrived at the art museum, Mrs. Gilson pointed out some of the art work in the courtyard of the museum. One of the paintings was considered “street art” and was immediately noticed by Tania and her friend Yalisha. The two girls walked all around the painting which was about five feet by five feet and stretched across an entire wall.
“This is really cool,” Tania commented. “I love the way the lines intersect. I think that this is a painting all about lines.”
“Me too,” Yalisha agreed. “However, there are angles here too. If you look, you can see that when the lines intersect they form different angles. For example, look at the small dark purple triangle and the light purple quadrilateral. The angles formed by those lines are exactly the same. Did you know that?”
“How can that be? One shape is so much larger than the other?” Tania asked puzzled.
How can it be? What it is about the relationship between lines that makes the angles the same or not the same? This lesson is all about angles and special pairs of angles. Keep Tania’s question in mind as you work through this lesson. At the end, see if you can figure out why Yalisha says that the angles are the same.
What You Will Learn
In this lesson you will learn to work with the following skills and concepts:
- Identify adjacent and vertical angles formed by two intersecting lines.
- Find measures of angles formed by two intersecting lines using known relationships and sufficient given information.
- Identify intersecting, parallel and perpendicular lines in a plane.
- Identify corresponding angles formed when a line intersects two other lines.
- Find measures of angles formed when a line intersects two other lines using known relationships and sufficient given information.
Teaching Time
I. Identify Adjacent and Vertical Angles Formed by Two Intersecting Lines
In this lesson we will look at the relationships among angles formed by intersecting lines. Some lines never intersect. Others do, and when they do, they form angles. Take a look at the intersecting lines below.
The angles formed by the two intersecting lines are numbered 1 through 4. In this lesson, we will learn how to find the measure of these angles, given the measure of any one of them.
Let’s look at the relationships formed between the angles created when two lines intersect.
Adjacent angles are angles that share the same vertex and one common side. If they combine to make a straight line, adjacent angles must add up to \begin{align*}180^\circ\end{align*}. Below, angles 1 and 2 are adjacent. Angles 3 and 4 are also adjacent. Adjacent angles can also be thought of as “next to” each other.
Can you see that angles 1 and 2, whatever their measurements are, will add up to \begin{align*}180^\circ\end{align*}? This is true for angles 3 and 4, because they also form a line. But that’s not all. Angles 1 and 4 also form a line. So do angles 2 and 3. These are also pairs of adjacent angles. Let’s see how this works with angle measurements.
The sum of each angle pair is \begin{align*}180^\circ\end{align*}. Using the vocabulary from the last section, you can also see that these angle pairs are supplementary.
This pattern of adjacent angles forms whenever two lines intersect. Notice that the two angles measuring \begin{align*}110^\circ\end{align*} are diagonal from each other, and the two angles measuring \begin{align*}70^\circ\end{align*} are diagonal from each other. This is the other special relationship among pairs of angles formed by intersecting lines.
What are angles on the diagonal called?
These angle pairs are called vertical angles. Vertical angles are always equal. Angles 1 and 3 above are vertical angles, and angles 2 and 4 are vertical angles.
These relationships always exist whenever any two lines intersect. Look carefully at the figures below. Understanding the four angles formed by intersecting lines is a very important concept in geometry.
In each figure, there are pairs of adjacent angles that add up to \begin{align*}180^\circ\end{align*} and pairs of vertical angles that are equal and opposite each other.
Now let’s practice recognizing adjacent and vertical angle pairs.
Example
Identify all of the pairs of adjacent angles and the two pairs of vertical angles in the figure below.
Hey, there aren’t any numbers! How are we supposed to know the measures of the four angles? Well, we actually don’t need to know them to answer the question. As we have said, adjacent and vertical relationships never change, no matter what the measures of the angles are. The pairs of adjacent angles will always form a straight line, and the pairs of vertical angles will always be opposite each other.
With this in mind, let’s look for the adjacent angles. Adjacent angles share a side and, in the case of intersecting lines, will together form a straight line. Which adjacent angles form line \begin{align*}n\end{align*}? Angles \begin{align*}Q\end{align*} and \begin{align*}R\end{align*} are next to each other and together make a straight angle along line \begin{align*}n\end{align*}. What about \begin{align*}T\end{align*} and \begin{align*}S\end{align*}? They also sit together along line \begin{align*}n\end{align*}. Both are adjacent pairs. Now let’s look at line \begin{align*}m\end{align*}. Which pairs of angles together make a straight angle along line \begin{align*}m\end{align*}? Angles \begin{align*}Q\end{align*} and \begin{align*}T\end{align*} do, and so do angles \begin{align*}S\end{align*} and \begin{align*}R\end{align*}. All four of these pairs are adjacent.
Now let’s look for the vertical angles. Remember, vertical angles are equal and opposite each other. Which angles are across from each other? Angles \begin{align*}Q\end{align*} and \begin{align*}S\end{align*} are, and we know that these have the same measure, whatever the measure is. Angles \begin{align*}T\end{align*} and \begin{align*}R\end{align*}, the small angles, are also opposite each other. Therefore they are the other pair of vertical angles.
Now that we can identify angle relationships, let’s look at figuring out angle measures.
II. Find Measures of Angles Formed by Two Intersecting Lines Using Known Relationships and Given Information
We can use what we have learned about adjacent and vertical angles to find the measure of an unknown angle formed by intersecting lines. We know that adjacent angles add up to \begin{align*}180^\circ\end{align*} and that vertical angles are equal. Therefore if we are given the measure of one angle, we can use its relationship to another angle to find the measure of the second angle. Let’s see how this works in a few examples.
Example Find the measure of angle \begin{align*}B\end{align*} below.
We know that one angle measures 50, and we want to find the measure of angle \begin{align*}B\end{align*}.
First we need to determine how these two angles are related. Is angle \begin{align*}B\end{align*} adjacent or vertical to the known angle? It is opposite, so these two angles are vertical angles, and we already know that vertical angles are always equal, so angle \begin{align*}B\end{align*} must also be \begin{align*}50^\circ\end{align*}.
Good for you. Let’s try another example.
Example Find the measure of \begin{align*}\angle Q\end{align*} below.
Again, we need to find how the known angle and the unknown angle are related. This time angle \begin{align*}Q\end{align*} is not opposite the known angle. It is adjacent, because together they form a straight line. What do we know about adjacent angles? They add up to \begin{align*}180^\circ\end{align*}. Therefore we can use the measure of the known angle to solve for angle \begin{align*}Q\end{align*}.
\begin{align*}138^\circ + \angle Q &= 180\\ \angle Q &= 180 - 138\\ \angle Q &= 42^\circ\end{align*} Angle \begin{align*}Q\end{align*} must be \begin{align*}42^\circ\end{align*}.
8D. Lesson Exercises
Find the missing angle measure by using the given information.
- Angle \begin{align*}A\end{align*} and Angle \begin{align*}B\end{align*} are adjacent angles. Angle \begin{align*}A\end{align*} is \begin{align*}85^\circ\end{align*}. What is the measure of Angle \begin{align*}B\end{align*}?
- Angle \begin{align*}C\end{align*} and Angle \begin{align*}D\end{align*} are vertical angles. If Angle \begin{align*}C\end{align*} is \begin{align*}55^\circ\end{align*}, what is the measure of Angle \begin{align*}D\end{align*}?
- Angle \begin{align*}E\end{align*} and Angle \begin{align*}F\end{align*} are adjacent angles. If Angle \begin{align*}E\end{align*} is \begin{align*}125^\circ\end{align*}, what is the measure of Angle \begin{align*}F\end{align*}?
Take a few minutes to check your answers with a friend.
Write down the definitions of vertical and adjacent angles in your notebook.
III. Identify Intersecting, Parallel and Perpendicular Lines in a Plane
What about the relationship between different types of lines? We have been working with intersecting lines and with the angle relationships that are formed by them, but there are other types of line relationships. Let’s look at them now.
What do we know about lines?
Lines exist in space. Two lines intersect when they cross each other. Because all lines are straight, intersecting lines can only cross each other once. Look at the examples below. Imagine the lines extend beyond the picture, on forever. Can you see how they will never cross more than once?
Two lines that form right angles when they intersect are perpendicular lines. All four angles formed by the perpendicular lines measure \begin{align*}90^\circ\end{align*}. We use a small box to show a right angle. If any of the four angles is marked with the small box, we know the lines are perpendicular. Take a look at the perpendicular lines below.
Some lines never intersect. We call these parallel lines. Parallel lines never cross, so they do not form any angles. Parallel lines look like railroad tracks; they are always the same distance apart, running next to each other.
One easy way to remember the difference between parallel and perpendicular lines is to look at the l’s in parallel. Parallel lines look exactly like the two l’s in their name!
Example
Identify whether the pairs of lines are parallel, perpendicular, or just intersecting.
As you look at each pair, first decide whether they intersect or if they might intersect if the lines were extended. If they do not and will not intersect, they must be parallel.
The lines in Figure 1 do not intersect. Let’s double check. Do they look like railroad tracks or the l’s in parallel? They do. The lines are always the same distance apart. No matter how far we extend them, they will never intersect. Figure 1 therefore shows parallel lines.
The rest of the pairs show intersecting lines. But which pairs are perpendicular? Remember, perpendicular lines form right angles when they intersect. If the two lines are too slanted, as in Figure 2, they cannot form right angles. Also, look for the little box that tells when an angle is a right angle. Let’s look at each of the pairs.
As we’ve said, the lines in Figure 2 are very slanted. They do not form perfect corners, or right angles, when they cross. They are intersecting lines, but not perpendicular lines.
The lines in Figure 3 do form right angles. The small box tells us that the lines definitely form right angles, so these are perpendicular lines.
Now let’s look at the lines in Figure 4. They do not intersect! But remember, lines continue in both directions forever. What would happen if we extend each line? Trace each with your finger. Will they cross if you extend them? They sure will. Now imagine what they will look like when they intersect, or draw a picture to help you. Would they meet at a slanted angle, or would they form right angles?
These lines do not form right angles, so they cannot be perpendicular. They are intersecting lines only.
Now let’s look at how lines affect corresponding angles.
IV. Identify Corresponding Angles Formed When a Line Intersects Two Other Lines
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 if the lines are parallel, 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.
Example 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.
V. Find Measures of Angles Using Known Relationships and Given Information
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.
Let’s look at the next example.
Example
Fill in the figure below with the angle measure for all of the angles shown.
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.
Real Life Example Completed
At the Art Museum
Have you been thinking about Tania’s question? Here is the original problem once again. Reread it and then write down your answer to her question. Are these two angles the same or aren’t they? Why or why not?'
When the students arrived at the art museum, Mrs. Gilson pointed out some of the art work in the courtyard of the museum. One of the paintings was considered “street art” and was immediately noticed by Tania and her friend Yalisha. The two girls walked all around the painting which was about five feet by five feet and stretched across an entire wall.
“This is really cool,” Tania commented. “I love the way the lines intersect. I think that this is a painting all about lines.”
“Me too,” Yalisha agreed. “However, there are angles here too. If you look, you can see that when the lines intersect they form different angles. For example, look at the small dark purple triangle and the light purple quadrilateral. The angles formed by those lines are exactly the same. Did you know that?”
“How can that be? One shape is so much larger than the other?” Tania asked puzzled.
Let’s think about Tania’s question. The size of the angles isn’t a function of whether or not the shapes are large or not. It has to do with the intersection of the lines. Remember how Tania commented that she thought that the painting was all about lines, well, here is where her point is valid.
First, think about what type of angles are formed by the two intersecting lines. We have vertical angles. Any time two lines intersect, the opposing angles formed by the intersection of those two lines is considered vertical angles. Vertical angles are congruent. Therefore, Yalisha’s comment is accurate.
Now look at the painting again. Can you find a pair of adjacent angles? Can you find another pair of vertical angles? Make a few notes in your notebook.
Vocabulary
- Adjacent Angles
- angles formed by intersecting lines that are supplementary and are next to each other. The sum of their angles is \begin{align*}180^\circ\end{align*}.
- Vertical Angles
- angles formed by intersecting lines that are on the diagonals. They have the same measure.
- Supplementary
- having a sum of \begin{align*}180^\circ\end{align*}.
- 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.
Technology Integration
Khan Academy Angles Formed Between Transversals and Parallel Lines
James Sousa, Angle Relationships and Triangles
Other Videos:
- http://www.teachertube.com/members/viewVideo.php?video_id=173907&title=VERTICAL_ANGLES – This is a video on vertical angles. It does not have sound and requires students to read the screen as the topic is demonstrated. You may need to register at the site to view this video.
- http://www.teachertube.com/members/viewVideo.php?video_id=174136&title=Angles – This video shows how angles show up in sports such as soccer. You may need to register at the site to view this video.
Time to Practice
Directions: Identify whether the lines below are parallel, perpendicular, or just intersecting.
1.
2.
3.
4.
5. Lines that will never intersect.
6. Lines that intersect at a \begin{align*}90^\circ\end{align*} angle.
7. Lines that cross at one point.
Directions: Tell whether the pairs of angles are adjacent or vertical. Then find the measure of the unknown angle. There will be two answers for each.
8.
9.
10.
11.
Directions: Find the measure of the unknown angle.
12.
13.
14.
15.
16.
Directions: Use what you have learned to answer each question true or false.
17. Adjacent angles are also supplementary angles.
18. Vertical angles are complementary angles.
19. If one adjacent angle is \begin{align*}100^\circ\end{align*}, then its angle pair is also \begin{align*}100^\circ\end{align*}.
20. Vertical angles have the same measure.
21. Vertical angles and corresponding angles are located in the same position.
22. You can have corresponding angles when you only have two intersecting lines.
23. Corresponding angles are in the same place given the intersection.
24. Parallel lines will never intersect.
25. Perpendicular lines intersect at a 90 degree angle.