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4.3: Parallel and Perpendicular Lines

Created by: CK-12

Lines and Angles

I. Section Objectives

  • Identify parallel lines, skew lines, and perpendicular lines
  • 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.

II. Multiple Intelligences

  • Teach half of the lesson, complete the walk around activity, and then finish the rest of the lesson.
  • Write the vocabulary on the board for the visual- spatial learners. Include the following.
    • parallel lines with symbol
    • perpendicular lines with symbol
    • parallel planes
    • skew lines
  • Have students walk around the room and make a list of all of the places where they can locate each type of lines.
  • When finished, allow time for students to share their findings.
  • Go on to the postulates. Break each postulate down into simple steps. This will assist visual- spatial learners and special needs students. Here are some suggestions.
  • Parallel Line Postulate
    • Given a line and a point not on that line
    • One line parallel to the given line goes through that point
  • Perpendicular Line Postulate
    • Given a line and a point not on that line
    • One line perpendicular to the line passes through that point
  • When working with transversals, use color to indicate the different angles in a diagram. Use color for definitions too. This will help students to keep things clear.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Before working on the section of the lesson on transversals, review the meaning of the following words: adjacent, vertical, interior, exterior, corresponding and consecutive.
  • Break down the transversal section. Use a diagram too.
  • Teach adjacent and vertical angles first.
  • Then teach interior, exterior and corresponding
  • Finally, teach alternate interior, alternate exterior and consecutive angles.

Parallel Lines and Transversals

I. Section 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.

II. Multiple Intelligences

  • This activity engages several of the intelligences, but also will demonstrate student understanding of the postulates and theorems in this lesson.
  • Divide the students into a group of four.
  • Assign each of the four students a different topic- Corresponding Angles Postulate, Alternate Interior Angles Theorem, Alternate Exterior Angles Theorem, Consecutive Interior Angles Theorem.
  • Let the students know that their assignment is to prepare a lesson and teach the other students in the group about their topic. They can use their notes, a diagram, real life examples, a poem, a song, whatever they would like to make the topic clear. The other students in the groups will let the “teacher” know what he/she did well and also offer suggestions to improve the presentation.
  • Intelligences- linguistic, logical- mathematical, musical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Review the angles formed by a transversal
  • Write all new postulates and theorems on the board/overhead. Request that the students copy this information down in their notebooks.
  • Be sure to place students in a group where they will be well supported by their peers. Some groups are more encouraging than others.

IV. Alternative Assessment

  • Create a rubric that covers the points that you want each student presentation to have.
  • Share this rubric with the students.
  • Observe students as they present their material.
  • Include group feedback as part of the rubric.
  • This could be used to calculate a quiz/classwork grade if necessary.

Proving Lines Parallel

I. Section Objectives

  • Identify and use the Converse of the Corresponding Angles Postulate.
  • Identify and use the Converse of Alternate Interior Angles Theorem.
  • Identify and use the Converse of Alternate Exterior Angles Theorem.
  • Identify and use the Converse of Consecutive Interior Angles Theorem.
  • Identify and use the Parallel Lines Property.

II. Multiple Intelligences

  • The best way to differentiate this lesson is to do so as part of a discussion. You want the students to make connections between the parallel lines, the transversal, and proving that the lines are parallel.
  • Demonstrate that it is possible to draw two lines and a transversal and have the lines not be parallel. This is where things being true or not true comes into the lesson.
  • The students are going to work with you as you work on the board/overhead. Begin by drawing two parallel lines and a transversal on the board.
  • Request that the students mirror this work at their seats. They will need paper, rules, pencils and protractors.
  • Then go through measuring each pair of angles.
  • Once this is finished, go though each postulate and theorem and demonstrate proving that the lines are parallel using the postulates and theorems.
  • Remind students that they are “proving” the accuracy of the statement.
  • Completing this lesson this way engages the following intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Review that what a conditional statement is and how to write the converse of a conditional statement.
  • Practice writing converse statements from conditional statements by using real life examples.
  • Be sure that the students understand this concept before moving on to the material in the lesson.
  • Review the Transitive Property.
  • Write all new terms on the board. Request that the students copy these notes in their notebooks.

IV. Alternative Assessment

  • The best way to assess student understanding is through observation.
  • Ask probing questions, allow plenty of think time, and listen carefully to student responses during the work of the lesson.

Slopes of Lines

I. Section Objectives

  • Identify and compute slope in the coordinate plane.
  • Use the relationship between slopes of parallel lines.
  • Use the relationship between slopes of perpendicular lines.
  • Plot a line on a coordinate plane using different methods.

II. Multiple Intelligences

  • For this lesson, be sure that students have grid paper, rulers and colored pencils.
  • Complete each of the exercises in the text as a whole class.
  • This will assist the students in practicing the constructions of lines and finding the slope of a line as you work with them.
  • It might even make sense to have a list of the ordered pairs in each example prepared ahead of time and not use the text at first. This way, you can go through each example with the students constructing lines and figuring slopes on their own without the answers presented in the text.
  • When you get to the parallel line section, you can introduce the theorem, and then have them construct the line and a line parallel to that given line.
  • When you get to the perpendicular line section, you can introduce the theorem and work with the students to draw in a line perpendicular to the line drawn.
  • Finally, work with students on using ordered pairs to graph different lines and to find the slopes of the lines. This is from the section on graphing strategies.
  • Intelligences: linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal, bodily- kinesthetic

III. Special Needs/Modifications

  • Review the following prior to beginning the lesson.
  • Drawing a coordinate grid
  • Labeling the coordinate grid
  • How to locate the x and y axis’
  • Review the origin as (0,0)
  • Review ordered pairs (x,y)
  • Review finding the reciprocal of a number/fraction
  • Write the two new theorems on the board and request that the students copy this information in their notebooks.

IV. Alternative Assessment

  • Alternative Assessment in this lesson can be done through observation. As the students work on the exercises, walk around the room and observe them as they work.
  • This is also a good time to notice students who need assistance.
  • If several students are needing assistance, consider allowing students to work in pairs.

Equations of Lines

I. Section Objectives

  • Identify and write equations in slope- intercept form.
  • Identify equations of parallel lines.
  • Identify equations of perpendicular lines.
  • Identify and write equations in standard form.

II. Multiple Intelligences

  • In this lesson, one way to differentiate this lesson is to break the material down into sections.
  • There is only one example per section, so you may want to include more than one so that the students have a chance to practice the concept before moving on to something new.
  • Include constructions whenever possible.
  • When working with slope- intercept form, show how in the equation y = mx + b, that m is the slope of the line and that the b is the y intercept.
  • Explain that the y intercept is where the line intersects with the y axis.
  • When figuring out which equation represents a line parallel to a line already graphed, provide students with these steps.
  1. Find the slope of the graphed line.
  2. Look at the equation choices
  3. Any equation with the same slope will be parallel to the graphed line.
  • To figure out the equation for a line perpendicular to a graphed line, follow these steps.
  1. Find the slope of the graphed line.
  2. Find the reciprocal of the slope.
  3. Any equation with the reciprocal as the slope is perpendicular to the line.
  • Intelligences- linguistic, logical- mathematical, visual- spatial

III. Special Needs/Modifications

  • Use the information above to scaffold this lesson for the students.
  • Ex 4- Standard Form- begin with a simpler example first.
  • Possible example 2y=6x+12
  • Write all steps on the board and request that the students copy those notes into their notebooks.

IV. Alternative Assessment

  • Observe students as they work. If there is a lot of confusion, review concepts more than once to be certain that students are following the material.

Perpendicular Lines

I. Section Objectives

  • Identify congruent linear pairs of angles.
  • Identify the angles formed by perpendicular intersecting lines.
  • Identify complementary adjacent angles.

II. Multiple Intelligences

  • Begin this lesson by reviewing the definition of perpendicular lines and the symbol for perpendicular lines.
  • List the description for each type of linear pair on the board.
  • Then ask the students to use the description to draw an example of the pair being described.
  • Example- Draw a linear pair of angles.
  • Next, show students how a triangle can be drawn into their diagram.
  • Request that they mark the right angle in the triangle.
  • Then assign one of the other angles in the triangle a measurement.
  • Request that students work to figure out the measurement of the missing angle.
  • Explain to students that because they know that the interior angles of a triangle add up to be 180^\circ that they can use this to figure out the missing angle of a triangle.
  • To expand upon this, ask the students to draw a diagram for a peer to solve. They need to include angle measurements too.
  • Then have them switch papers and solve each other’s problems.
  • Intelligences- linguistic, logical- mathematical, visual- spatial, interpersonal, intrapersonal

III. Special Needs/Modifications

  • Review the following vocabulary prior to beginning the lesson.
  • Congruent
  • Perpendicular
  • Complementary
  • Supplementary
  • Vertical angles
  • Adjacent angles
  • Linear pair

IV. Alternative Assessment

  • Walk around the room as students work.
  • Collect the problems/diagrams that they created for a peer to solve.
  • Use these as a classwork grade or to assess student understanding.

Perpendicular Transversals

I. Section Objectives

  • Identify the implications of perpendicular transversals on parallel lines.
  • Identify the converse theorems involving perpendicular transversals and parallel lines.
  • Understand and use the distance between parallel lines.

II. Multiple Intelligences

  • Prior to teaching this lesson, provide students with another example. Request that they draw it out and then discover the 90^\circ angles for themselves. Then move on to the text.
  • Having the students engage right away and draw conclusions about the material will help to reaffirm the new information in their minds.
  • Assist students in completing as many constructions as they can throughout this lesson. Drawing out the examples will expand student understanding.
  • Have students practice one or two more examples of finding the distance between straight parallel lines before moving on to slanted parallel lines.
  • When you begin working on slanted parallel lines, list these steps on the board. Request that the students copy these steps into their notebooks.
  1. Choose two points on one of the lines.
  2. Use those points to find the slope of the line.
  3. Take the slope and find the slope of the segment perpendicular to the line- the opposite of the reciprocal.
  4. Select a point on the line and use the slope to draw a segment perpendicular to the line.
  5. Take the coordinates of where this segment intersects both lines.
  6. Use the distance formula and these coordinates to find the distance between the parallel lines.

Writing out the steps in this fashion assists students with verbally seeing something, writing it down, and talking about it. Students will tend to remember the information better.

III. Special Needs/Modifications

  • Review the following vocabulary.
  • Perpendicular lines
  • Parallel lines
  • Transversals
  • Corresponding angles
  • Alternate interior angles
  • Alternate exterior angles
  • Consecutive interior angles
  • Review converse and remind students that this is working backwards in a way.

Non- Euclidean Geometry

I. Section Objectives

  • Understand non- Euclidean geometry concepts.
  • Find taxicab distances.
  • Identify and understand taxicab circles.
  • Identify and understand taxicab midpoints.

II. Multiple Intelligences

  • This is a great lesson to differentiate because taxicab geometry lends itself to creative problems.
  • Go through the material in this lesson, and then divide students into groups. Assign each group either a taxicab distance or a taxicab circle.
  • The students then work to write a problem to determine a taxicab distance or a taxicab circle.
  • After writing the problem, the students need to draw out a diagram that shows their problem and the solution.
  • Finally, the students act out the problem and the solution to the problem. They can use desks or different markers in the room to represent the area or “blocks” that they are working with.
  • Be sure that students understand the directions and whether or not they will be graded on their work.
  • If you are using a rubric for grading, share it with the students prior to their work.
  • Intelligences- linguistic, logical- mathematical, bodily- kinesthetic, visual- spatial, interpersonal, intrapersonal.

III. Special Needs/Modifications

  • Review circles including radius
  • Write a definition for taxicab geometry, taxicab circles, taxicab distance and taxicab midpoint on the board.
  • Request that students copy the information down in their notebooks.

IV. Alternative Assessment

  • This is a great opportunity to design a rubric for the activity.
  • The rubric can be divided into three sections: the problem itself, the diagram and the skit.
  • Students have many opportunities to excel in this assignment because of all of the different ways to participate.
  • This could be used as a quiz grade or a classwork grade.

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CK.MAT.ENG.TE.1.Geometry.4.3

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