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2.3: Parallel and Perpendicular Lines

Created by: CK-12

Lines and Angles

Marking the Diagram – Sometimes students confuse the marks for parallel and congruent. When introduction them to the arrows that represent parallel lines, review the ticks that represent congruent segments. Seeing the two at the same time helps avoid confusion.

When given the information that two lines or segments are perpendicular, students don’t always immediately see how to mark the diagram accordingly. They need to use the definition of perpendicular and mark one of the right angles created by the lines with a box.

Symbol Update – Students should be keeping a list of symbols and how they will be used in this class in their notebooks. Remind them to update this page with the symbols for parallel and perpendicular.

Construction – The parallel and perpendicular line postulates are used in construction. Constructing parallel and perpendicular lines with a compass and straightedge is a good way to give students kinesthetic experience with these concepts. Construction can also be done with computer software. To construct a parallel or perpendicular line the student will select the line they want the new line to be parallel or perpendicular to, and the point they want the new line to pass through, and chose construct. The way the programs have the students select the line and then the point reinforces the postulates.

Additional Exercises:

1. Write a two-column proof of the following conditional statement.

If \overline{AB} is perpendicular to \overline{BC}, triangle ABC is a right triangle.

Answer:

Statement Reason
\overline{AB} is perpendicular to \overline{BC} Given.
<B is right Definition of perpendicular.
triangle ABC is a right triangle Definition of right triangle.

Parallel Lines and Transversals

The Parallel Hypothesis – So far seven different pairs of angles that may be supplementary or congruent have been introduced. All seven of these pairs are used in the situation where two lines are being crossed by a transversal forming eight angles. Some of these pairs require the two lines to be parallel and some do not. Students sometimes get these confuse on when they need parallel lines to apply a postulate or theorem, and if a specific pair is congruent or supplementary. A chart like the one below will help them sort it out.

Type of Angle Pair Relationship
Do Not Require Parallel Lines Linear Pairs Supplementary
Vertical Angles Congruent
Parallel Lines Required Corresponding Angles Congruent
Alternate Interior Angles Congruent
Alternate Exterior Angles Congruent
Consecutive Interior Angles Supplementary
Consecutive Exterior Angles Supplementary

Patty Paper Activity – When two lines are intersected by a transversal eight angles are formed in two sets of four. When the lines are parallel, the two sets of four angles are exactly the same. To help students see this relationship, have them darken a set of parallel lines on their binder paper a few inches apart and draw a transversal through the parallel lines. Now they should trace one set of four angles on some thin paper (tracing paper or patty paper). When they slide the set of four angles along the transversal they will coincide with the other set of four angles. Have them try the same thing with a set of lines that are not parallel. This will help students find missing angle measures quickly and remember when they can transfer numbers down the transversal. It does not help them learn the names of the different pairs of angles which in important for communicating with others about mathematical concepts and for writing proofs.

Additional Exercises:

1. One angle of a linear pair has a measure twice as large as the other angle. What are the two angle measures?

Answer:

 x + 2x & = 180 && \text{The angles measure}\ 60\ \text{degrees and}\ 120\ \text{degrees} \\x & = 60

Proving Lines Parallel

When to Use the Converse – It takes some experience before most students truly understand the difference between a statement and its converse. They will be able to write and recognize the converse of a statement, but then will have a hard time deciding which one applies in a specific situation. Tell them when you know the lines are parallel and are looking for angles, you are using the original statements; when you are trying to decide if the lines are parallel or not, you are using the converse.

Additional Exercise:

1. Prove the Converse of the Alternate Exterior Angle Theorem.

Answer: Refer to the image used to prove the Converse of the Alternate Interior Angle Theorem in the text.

Statement Reason
\angle ABC \cong \angle HFE Given
\angle HFE \cong \angle GFB Vertical Angles Theorem
\angle ABC \cong \angle GFB Transitive Property of Angle Congruence
\overleftrightarrow{AD} is parallel to \overleftrightarrow{GE} Converse of the Corresponding Angles Postulate.

The Converse of the Alternate Exterior Angle Theorem could also be proved using the Converse of the Alternate Interior Angle Theorem. This would demonstrate to the students that once a theorem has been proved it, can be used in the proof of other theorems. It demonstrates the building block nature of math.

Statement Reason
\angle ABC \cong \angle HFE Given

\angle HFE \cong \angle GFB

\angle DBF \cong \angle GFB

Vertical Angles Theorem
\angle DBF \cong \angle GFB Transitive Property of Angle Congruence
\overleftrightarrow{AD} is parallel to \overleftrightarrow{GE} Converse of the Alternate Interior Angles Theorem.

Proving the theorem in several ways gives students a chance to practice with the concepts and their proof writing skills. Similar proofs can be assigned for the other theorems in this section.

Slopes

Order of Subtraction – When calculating the slope of a line using two points it is important to keep straight which point was made point one and which one was point two. It does not matter how these labels are assigned, but the order of subtraction has to stay the say in the numerator and the denominator of the slope ratio. If students switch the order they will get the opposite of the correct answer. If they have a graph of the line, ask them to compare the sign of the slope to the direction of the line. Is the line increasing or decreasing? Does that match the slope?

Graphing Lines with Integer Slopes – The slope of a line is the ratio of two numbers. When students are asked to graph a line with an integer slope they often fail to realize what and where the second number is. Frequently they will make the “run” of the line zero and graph a vertical line. It is helpful to have them write the integer that is the slope, as a ratio over one, before then do any graphing. Really, they only need to do this a few times on paper before they are able to graph the lines correctly. They will begin to see the ratio correctly in their heads.

Zero or Undefined – Students need to make these associations:

Zero in numerator – slope is zero – line is horizontal

Zero in denominator – slope is undefended – line is vertical

They frequently switch these around. After the relationships are explained in class, remind them frequently, maybe have a poster up in the room or write the relationship on a corner of the board that does not get erased.

Use Graph Paper – Making a connection between the numbers that describe a line and the line itself is an important skill. Requiring that the students use graph paper encourages them to make nice, thoughtful graphs, and helps them make this connection.

Additional Exercises:

1. Find the slope of the line that is perpendicular to the line passing through the points (5, -7), and (-2, -3).

Answer: The slope of the line passing through the given points is \frac{-4}{7}, so the line perpendicular to this line has the slope \frac{7}{4}.

Equations of Lines

The y-axis is Vertical – When using the slope-intercept form to graph a line or write an equation, it is common for students to use the x-intercept instead of the y-intercept. Remind them that they want to use the vertical axis, y-intercept, to begin the graph. Requiring that the y-intercept be written as a point, say (0,3) instead of just 3, helps to alleviate this problem.

Where’s the Slope – Students are quickly able to identify the slope as the coefficient of the x-variable when a line is in slope-intercept form, unfortunately they sometimes extend this to standard form. Remind the students that if the equation of a line is in standard form, or any other form, they must first algebraically convert it to slope-intercept form before they can easily read off the slope.

Key Exercises:

1. Write the equation 3x+5y=10 in slope-intercept form.

Answer: y=-\frac{3}{5}x+2

2. What is the slope of the line 2x-3y=7?

Answer: \frac{3}{2}

3. Are the lines below parallel, perpendicular, or neither?

6x+ 4y & =7 \\6x-4y & = 7

Answer: These lines are neither parallel nor perpendicular.

Why Use Standard Form – The slope-intercept form of the line holds so much valuable information about the graph of a line, that students probably won’t understand why any other form would ever be used. Mention to them that standard form is convenient when putting equations into matrices, something they will be doing in their second year of algebra, to motivate them to learn and remember the standard form.

Perpendicular Lines

Complementary, Supplementary, or Congruent – When finding angle measures students generally need to decide between three possible relationships: complementary, supplementary, and congruent. A good way for them to practice with these and review their equation solving skills, is to assign variable expressions to angle measures, state the relationship of the angles, and have the students use this information to write an equation that when solved will lead to a numerical measurement for the angle.

Key Exercise:

1. Two vertical angles have measures 2x-30^\circ and x+60^\circ.

Set-up and solve an equation to find x. Then find the measures of the angles.

Answer:

2x-30^\circ & = x+60^\circ && \text{Both angles have a measure of}\ 150 \ \text{degrees}.\\x & = 90^\circ

2. The outer rays of two adjacent angle with measures 4x+10^\circ and 5x-10^\circ are perpendicular. Find the measures of each angle.

Answer:

5x-10+4x+10 & = 90 && \text{The angles have measures of}\ 50 \ \text{degrees  and}\  40\ \text{degrees}.\\x & = 10

3. The angles of a linear pair have measures 3x+45^\circ and 2x+35^\circ. Find the measure of each angle.

Answer:

2x+35+3x+45 & =180 && \text{The angles have measures of}\ 105 \ \text{degrees and}\ 75 \ \text{degrees}.\\x & =20

Encourage students to take the time to write out and solve the equation neatly. This process helps them avoid errors. Many times students will find the value of x, and then stop without plugging in the value to the expression for the angle measures. Have the students verify that their final answers are angle measures that have the desired relationship.

Additional Exercise:

1. Perpendicular lines form an angle with measure 8x+10^\circ. What is the value of x?

Answer:

8x+10^\circ & =90^\circ \\x & =10^\circ

Perpendicular Transversals

The Perpendicular Distance – In theory, measuring along a perpendicular line makes sense to the students, but in practice, when lining up the ruler or deciding which points to put in the distance formula, there are many distractions. Students can evaluate their decision by taking a second look to see if the path they chose was the shortest one possible.

Multi-Step Procedures – When working on an exercise that requires many different steps, like the last problem in this section, students sometimes become lost in the process or overwhelmed before they begin. A good way to ground students, and help them move through the problem, is to create, or have them create, a To-do list. Writing out the steps that need to be completed will help them understand the process, give them a sense of satisfaction as the check off parts they have completed, and help them organize their work. Creating the list could be a good group activity.

Where to Measure? – Now that the students know to measure along a line that is perpendicular to both parallel lines, they might wonder where along the lines to measure. When working on a coordinate plane it is best to start with a point that has integer coordinates, just to keep the problem simple and accurate. They will get the same distance no matter where they measure though. An alternate definition of parallel lines is two lines that are a constant distance apart.

Addition Exercises:

1. Prove the Converse of the Perpendicular Transversal Theorem.

Answer: Refer to the figure at the top of page 178, at the beginning of this lesson.

Statement Reason
\overleftrightarrow{KN} is perpendicular to \overleftrightarrow{QT} Given
\overleftrightarrow{OR} is perpendicular to \overleftrightarrow{QT} Given
\angle QPO is right Definition of Perpendicular Lines
\angle PST is right Definition of Perpendicular Lines
\angle QPO \cong \angle PST Right Angle Theorem
\overleftrightarrow{OR} is parallel to \overleftrightarrow{KN} Converse of Corresponding Angle Postulate

Non-Euclidean Geometry

Separate Worlds – The geometry presented in this section is completely separate from the geometry in the rest of the text. The study of non-Euclidean geometry is excellent for developing critical thinking skills. It also demonstrates to the students what an influential role postulates play and how important it is to carefully evaluate them before accepting them as true. This section is best used for enrichment and should be treated differently from the other sections. If the students attempt to memorize the postulates in this section it may compromise their ability to recall analogous postulates of Euclidean Geometry. Exploring taxicab geometry is a wonderful way to spend a day in class, but it is not something that has to be included on tests. This is a decision that the instruction can make based on the ability of the students in a particular class.

Projects – This section opens the door to many possible projects that students can complete as part of the class or for extra credit. More advanced students in particular will have the ability and interest to explore the topic of Non-Euclidean geometry independently. Topics can include further exploration of taxicab geometry, other types of Non-Euclidean geometry, like spherical geometry, or research into the mathematician who developed these fields. This may make a good group project, where each group presents its findings to the class.

Encourage Creativity - Have students write their own problems involving taxicab geometry. This type of geometry lends itself to application and story problems. Students can be creative and funny. They will enjoy sharing problems with their classmates and solving each other’s challenges. Writing word application helps students solve similar exercises. When formulating their question and deciding what information to give and how to give it, they become more aware of the structure of a word problem. If the students are enjoying this line of study and there is time, they may create their own type of geometry by setting up a system of postulates.

Abstraction and Modeling– This section briefly addresses the fact that mathematics is an abstraction and that it usually needs to be modified before it can be helpful in applications to the world in which we live. This is an important concept applicable to all areas of mathematics that is easily seen while studying geometry. This knowledge will help students understand why math is useful and how they will benefit from what they are learning in this class.

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CK.MAT.ENG.TE.1.Geometry.2.3

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