Review Questions
Use the diagram to answer questions 1-3.
Given:
Fill in the blanks.
_________
_________
_________
Fill in the reasons in the following proof. Given: and
Prove:
Statement Reason
and
a.____
and are right angles
b.____
and
c.____
d.____
e.____
and are complementary , and are complementary
f.____
g.___
Which of the following statements must be true? Answer Yes or No.
The following diagram shows a ray of light that is reflected from a mirror. The dashed segment is perpendicular to the mirror. .
is called the angle of incidence; is called the angle of reflection. Explain how you know that the angle of incidence is congruent to the angle of reflection.
Review Answers
Given
Definition of Perpendicular Segments
Angle Addition Postulate
Definition of Right Angle
Substitution (Transitive Property of Equality)
Definition of Complementary Angles
Complements of the Same Angle are Congruent
No
Yes
No
and are complementary; and are complementary . because they are complements of congruent angles and .
Chapter 3: Parallel and Perpendicular Lines
Lines and Angles
Learning Objectives
Identify parallel lines, skew lines, and parallel planes.
Know the statement of and use the Parallel Line Postulate.
Know the statement of and use the Perpendicular Line Postulate.
Identify angles made by transversals.
Introduction
In this chapter, you will explore the different types of relationships formed with parallel and perpendicular lines and planes. There are many different ways to understand the angles formed, and a number of tricks to find missing values and measurements. Though the concepts of parallel and perpendicular lines might seem complicated, they are present in our everyday life. Roads are often parallel or perpendicular, as are crucial elements in construction, such as the walls of a room. Remember that every theorem and postulate in this chapter can be useful in practical applications.
Parallel and Perpendicular Lines and Planes, and Skew Lines
Parallel lines are two or more lines that lie in the same plane and never intersect.
We use the symbol for parallel, so to describe the figure above we would write . When we draw a pair of parallel lines, we use an arrow mark to show that the lines are parallel. Just like with congruent segments, if there are two (or more) pairs of parallel lines, we use one arrow for one pair and two (or more) arrows for the other pair.
Perpendicular lines intersect at a right angle. They form a angle. This intersection is usually shown by a small square box in the angle.
The symbol is used to show that two lines, segments, or rays are perpendicular. In the preceding picture, we could write . (Note that is a ray while is a line.)
Note that although "parallel" and "perpendicular" are defined in terms of lines, the same definitions apply to rays and segments with the minor adjustment that two segments or rays are parallel (perpendicular) if the lines that contain the segments or rays are parallel (perpendicular).
Example 1
Which roads are parallel and which are perpendicular on the map below?
The first step is to remember the definitions or parallel and perpendicular lines. Parallel lines lie in the same plane but will never intersect. Perpendicular lines intersect at a right angle. All of the roads on this map lie in the same plane, and Rose Avenue and George Street never intersect. So, they are parallel roads. Henry Street intersects both Rose Avenue and George Street at a right angle, so it is perpendicular to those roads.
Planes can be parallel and perpendicular just like lines. Remember that a plane is a two-dimensional surface that extends infinitely in all directions. If planes are parallel, they will never intersect. If they are perpendicular, they will intersect at a right angle.
If you think about a table, the top of the table and the floor below it are usually in parallel planes.
The other of relationship you need to understand is skew lines. Skew lines are lines that are in different planes, and never intersect. Segments and rays can also be skew. In the cube shown below segment and segment are skew. Can you name other pairs of skew segments in this diagram? (How many pairs of skew segments are there in all?)
Example 2
What is the relationship between the front and side of the building in the picture below?
The planes that are represented by the front and side of the building above intersect at the corner. The corner appears to be a right angle so the planes are perpendicular.
Parallel Line Postulate
As you already know, there are many different postulates and theorems relating to geometry. It is important for you to maintain a list of these ideas as they are presented throughout these chapters. One of the postulates that involves lines and planes is called the Parallel Line Postulate.
Parallel Postulate: Given a line and a point not on the line, there is exactly one line parallel to the given line that goes through that point. Look at the following diagram to see this illustrated.
Line in the diagram above is near point . If you want to draw a line that is parallel to that goes through point there is only one option. Think of lines that are parallel to as different latitude, like on a map. They can be drawn anywhere above and below line , but only one will travel through point .
Example 3
Draw a line through point that is parallel to line .
Remember that there are many different lines that could be parallel to line .
There can only be one line parallel to that travels through point . This line is drawn below.
Perpendicular Line Postulate
Another postulate that is relevant to these scenarios is the Perpendicular Line Postulate.
Perpendicular Line Postulate: Given a line and a point not on the line, there is exactly one line perpendicular to the given line that passes through the given point.
This postulate is very similar to the Parallel Line Postulate, but deals with perpendicular lines. Remember that perpendicular lines intersect at a right angle. So, as in the diagram below, there is only one line that can pass through point while being perpendicular to line .
Example 4
Draw a line through point that is perpendicular to line .
Remember that there can only be one line perpendicular to that travels through point . This line is drawn below.
Angles and Transversals
Many math problems involve the intersection of three or more lines. Examine the diagram below.
In the diagram, lines and are crossed by line . We have quite a bit of vocabulary to describe this situation:
Line is called a transversal because it intersects two other lines ( and ). The intersection of line with and forms eight angles as shown.
The area between lines and is called the interior of the two lines. The area not between lines and is called the exterior.
Angles and are called adjacent angles because they share a side and do not overlap. There are many pairs of adjacent angles in this diagram, including and and , and and .
and are vertical angles. They are nonadjacent angles made by the intersection of two lines. Other pairs of vertical angles in this diagram are and and , and and .
Corresponding angles are in the same position relative to both lines crossed by the transversal. is on the upper left corner of the intersection of lines and . is on the upper left corner of the intersection of lines and . So we say that and are corresponding angles.
and are called alt
ernate interior angles. They are in the interior region of the lines and and are on opposite sides of the transversal.
Similarly, and are alternate exterior angles because they are on opposite sides of the transversal, and in the exterior of the region between and .
Finally, and are consecutive interior angles. They are on the interior of the region between lines and and are next to each other. and are also consecutive interior angles.
Example 5
List all pairs of alternate angles in the diagram below.
There are two types of alternate angles—alternate interior angles and alternate exterior angles. As you need to list them both, begin with the alternate interior angles.
Alternate interior angles are on the interior region of the two lines crossed by the transversal, so that would include angles and Alternate angles are on opposite sides of the transversal, . So, the two pairs of alternate interior angles are , and and .
Alternate exterior angles are on the exterior region of the two lines crossed by the transversal, so that would include angles and Alternate angles are on opposite sides of the transversal, . So, the two pairs of alternate exterior angles are , and and .
Lesson Summary
In this lesson, we explored how to work with different types of lines, angles and planes. Specifically, we have learned:
How to identify parallel lines, skew lines, and parallel planes.
How to identify and use the Parallel Line Postulate.
How to identify and use the Perpendicular Line Postulate.
How to identify angles and transversals of many types.
These will help you solve many different types of problems. Always be on the lookout for new and interesting ways to examine the relationship between lines, planes, and angles.
Points to Consider
Parallel planes are two planes that do not intersect. Parallel lines must be in the same plane and they do not intersect. If more than two lines intersect at the same point and they are perpendicular, then they cannot be in same plane (e.g., the , , and axes are all perpendicular). However, if just two lines are perpendicular, then there is a plane that contains those two lines.
As you move on in your studies of parallel and perpendicular lines you will usually be working in one plane. This is often assumed in geometry problems. However, you must be careful about instances where you are working with multiple planes in space. Generally in three-dimensional space parallel and perpendicular lines are more challenging to work with.
Review Questions
Solve each problem.
Imagine a line going through each branch of the tree below (see the red lines in the image). What term best describes the two branches with lines in the tree pictured below?
How many lines can be drawn through point that will be parallel to line
Which of the following best describes skew lines? They lie in the same plane but do not intersect.
They intersect, but not at a right angle.
They lie in different planes and never intersect.
They intersect at a right angle.
Are the sides of the Transamerica Pyramid building in San Francisco parallel?
How many lines can be drawn through point that will be perpendicular to line
Which of the following best describes parallel lines? They lie in the same plane but do not intersect.
They intersect, but not at a right angle.
They lie in different planes and never intersect.
They intersect at a right angle.
Draw five parallel lines in the plane. How many regions is the plane divided into by these five lines?
If you draw parallel lines in the plane, how many regions will the plane be divided into?
The diagram below shows two lines cut by a transversal. Use this diagram to answer questions 9 and 10.
What term best describes the relationship between angles 1 and 5? Consecutive interior
Alternate exterior
Alternate interior
Corresponding
What term best describes angles 7 and 8? Linear pair
Alternate exterior
Alternate interior
Corresponding
Review Answers
Skew
One
C
No
One
A
Five parallel lines divide the plane into six regions
parallel lines divide the plane into regions
D
A
Parallel Lines and Transversals
Learning Objectives
Identify angles formed by two parallel lines and a non-perpendicular transversal.
Identify and use the Corresponding Angles Postulate.
Identify and use the Alternate Interior Angles Theorem.
Identify and use the Alternate Exterior Angles Theorem.
Identify and use the Consecutive Interior Angles Theorem.
Introduction
In the last lesson, you learned to identify different categories of angles formed by intersecting lines. This lesson builds on that knowledge by identifying the mathematical relationships inherent within these categories.
Parallel Lines with a Transversal—Review of Terms
As a quick review, it is helpful to practice identifying different categories of angles.
Example 1
In the diagram below, two vertical parallel lines are cut by a transversal.
Identify the pairs of corresponding angles, alternate interior angles, alternate exterior angles, and consecutive interior angles.
Corresponding angles: Corresponding angles are formed on different lines, but in the same relative position to the transversal—in other words, they face the same direction. There are four pairs of corresponding angles in this diagram— and and and , and and .
Alternate interior angles: These angles are on the interior of the lines crossed by the transversal and are on opposite sides of the transversal. There are two pairs of alternate interior angles in this diagram— and , and and .
Alternate exterior angles: These are on the exterior of the lines crossed by the transversal and are on opposite sides of the transversal. There are two pairs of alternate exterior angles in this diagram— and , and and .
Consecutive interior angles: Consecutive interior angles are in the interior region of the lines crossed by the transversal, and are on the same side of the transversal. There are two pairs of consecutive interior angles in this diagram— and and and .
Corresponding Angles Postulate
By now you have had lots of practice and should be able to easily identify relationships between angles.
Corresponding Angles Postulate: If the lines crossed by a transversal are parallel, then corresponding angles will be congruent. Examine the following diagram.
You already know that and are corresponding angles because they are formed by two lines crossed by a transversal and have the same relative placement next to the transversal. The Corresponding Angles postulate says that because the lines are parallel to each other, the corresponding angles will be congruent.
Example 2
In the diagram below, lines and are parallel. What is the measure of ?
Because lines and are parallel, the angle and are corresponding angles, we know by the Corresponding Angles Postulate that they are congruent. Therefore, .
Alternate Interior Angles Theorem
Now that you know the Corresponding Angles Postulate, you can use it to derive the relationships between all other angles formed when two lines are crossed by a transversal. Examine the angles formed below.
If you know that the measure of is you can find the measurement of all the other angles. For example, and must be supplementary (sum to ) because together they are a linear pair (we are using the Linear Pair Postulate here). So, to find , subtract from
So, . Knowing that and are also supplementary means that since . If , then must be because and are also supplementary. Notice t
hat (they both measure ) and (both measure ). These angles are called vertical angles. Vertical angles are on opposite sides of intersecting lines, and will always be congruent by the Vertical Angles Theorem, which we proved in an earlier chapter. Using this information, you can now deduce the relationship between alternate interior angles.
Example 3
Lines and in the diagram below are parallel. What are the measures of angles and ?
In this problem, you need to find the angle measures of two alternate interior angles given an exterior angle. Use what you know. There is one angle that measures Angle corresponds to the angle. So by the Corresponding Angles Postulate, .
Now, because is made by the same intersecting lines and is opposite the angle, these two angles are vertical angles. Since you already learned that vertical angles are congruent, we conclude . Finally, compare angles and . They both measure so they are congruent. This will be true any time two parallel lines are cut by a transversal.
CK-12 Geometry Page 9