Geometry (A): READ: Lines and Angles Reading

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 $\overleftrightarrow{MN} \| \overleftrightarrow{CD}$ . 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 $90^\circ$ angle. This intersection is usually shown by a small square box in the $90^\circ$ angle.

The symbol $\perp$ is used to show that two lines, segments, or rays are perpendicular. In the preceding picture, we could write $\overrightarrow {BA} \perp \overleftrightarrow{BC}$ . (Note that $\overrightarrow {BA}$ is a ray while $\overleftrightarrow{BC}$ 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 $\overline{AB}$ and segment $\overline{CG}$ 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 $(90^\circ),$ 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 $m$ in the diagram above is near point $D$ . If you want to draw a line that is parallel to $m$ that goes through point $D$ there is only one option. Think of lines that are parallel to $m$ as different latitude, like on a map. They can be drawn anywhere above and below line $m$ , but only one will travel through point $D$ .

Example 3

Draw a line through point $R$ that is parallel to line $s$ .

Remember that there are many different lines that could be parallel to line $s$ .

There can only be one line parallel to $s$ that travels through point $R$ . 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 $(90^\circ)$ angle. So, as in the diagram below, there is only one line that can pass through point $B$ while being perpendicular to line $a$ .

Example 4

Draw a line through point $D$ that is perpendicular to line $e$ .

Remember that there can only be one line perpendicular to $e$ that travels through point $D$ . 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 $g$ and $h$ are crossed by line $l$ . We have quite a bit of vocabulary to describe this situation:

Line $l$ is called a transversal because it intersects two other lines ( $g$ and $h$ ). The intersection of line $l$ with $g$ and $h$ forms eight angles as shown.
The area between lines $g$ and $h$ is called the interior of the two lines. The area not between lines $g$ and $h$ is called the exterior.
Angles $\angle{1}$ and $\angle{2}$ are called adjacent angles because they share a side and do not overlap. There are many pairs of adjacent angles in this diagram, including $\angle{2}$ and $\angle{3}, \angle{4}$ and $\angle{7}$ , and $\angle{8}$ and $\angle{1}$ .
$\angle{1}$ and $\angle{3}$ are vertical angles. They are nonadjacent angles made by the intersection of two lines. Other pairs of vertical angles in this diagram are $\angle{2}$ and $\angle{8}, \angle{4}$ and $\angle{6}$ , and $\angle{5}$ and $\angle{7}$ .
Corresponding angles are in the same position relative to both lines crossed by the transversal. $\angle{1}$ is on the upper left corner of the intersection of lines $g$ and $l$ . $\angle{7}$ is on the upper left corner of the intersection of lines $h$ and $l$ . So we say that $\angle {1}$ and $\angle {7}$ are corresponding angles.
$\angle{3}$ and $\angle{7}$ are called alternate interior angles. They are in the interior region of the lines $g$ and $h$ and are on opposite sides of the transversal.
Similarly, $\angle{2}$ and $\angle{6}$ are alternate exterior angles because they are on opposite sides of the transversal, and in the exterior of the region between $g$ and $h$ .
Finally, $\angle{3}$ and $\angle{4}$ are consecutive interior angles. They are on the interior of the region between lines $g$ and $h$ and are next to each other. $\angle{8}$ and $\angle{7}$ 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 $3, 4, 5,$ and $6.$ Alternate angles are on opposite sides of the transversal, $z$ . So, the two pairs of alternate interior angles are $\angle{3} \And \angle{5}$ , and $\angle{4}$ and $\angle{6}$ .

Alternate exterior angles are on the exterior region of the two lines crossed by the transversal, so that would include angles $1, 2, 8,$ and $7.$ Alternate angles are on opposite sides of the transversal, $z$ . So, the two pairs of alternate exterior angles are $\angle{2} \And \angle{8}$ , and $\angle{1}$ and $\angle{7}$ .

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 $x-$ , $y-$ , and $z-$ 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.

Last modified: Monday, May 10, 2010, 2:19 PM