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Home , Parallel (geometry)

Educator’s Guide to Lines and Transversals

Overview: Students will look at the relationship between parallel lines and the that runs through the lines.

Grades and Subject Areas: High School

Objective: Students will explore and deepen their understanding of the relationship of the angles formed by parallel lines and a transversal.

I can statements: I can name the special angle relationships formed by parallel lines and a transversal. (Corresponding, Alternate Interior, Alternate Exterior, Same-Side Interior or Consecutive Interior) I can construct lines, points, and measure angles using Geometer’s Sketchpad. I can demonstrate my knowledge of vertical angles and linear pair of angles.

Curriculum Connections/Alaska Standards: Alaska Math GLE [10] G-1 identifying, analyzing, comparing, or using properties of figures: Supplementary, complementary or vertical angles Angles created by parallel lines with a transversal

Created by Don Benn 1 October 15, 2011

ISTE Student Standards:

4. Critical Thinking, Problem-Solving & Decision-Making Students use critical thinking skills to plan and conduct research, manage projects, solve problems and make informed decisions using appropriate digital tools and resources. Students: c. - Collect and analyze data to identify solutions and/or make informed decisions. 6. Technology Operations and Concepts Students demonstrate a sound understanding of technology concepts, systems and operations. Students: a. – Troubleshoot systems and applications

ISTE Teacher Standards: 4. TEACHING, LEARNING, AND THE CURRICULUM Teachers implement curriculum plans that include methods and strategies for applying technology to maximize student learning. Teachers: A. Facilitate technology-enhanced experiences that address content standards and student technology standards. B. Use technology to support learner-centered strategies that address the diverse needs of students. D. Manage student learning activities in a technology-enhanced environment.

Hardware and Software Needs: Students and teachers will need access to computers that have the program Geometer’s Sketchpad.

Resources: Documents needed: Attached “Parallel Lines and Transversals” and “Vertical Lines”

Video Tutorials: http://www.atomiclearning.com/k12/geomsketch_mac http://www.atomiclearning.com/k12/geomsketch_pc

Web resources: http://sketchexchange.keypress.com/

Created by Don Benn 2 October 15, 2011

Parallel Lines and Transversals

Lesson Plan

Prep Time: 30 - 45 Minutes

Prior to Lesson: Reserve a computer lab that has Geometer’s Sketchpad. Print and copy the handouts “Parallel Lines and Transversals and “Vertical Lines.”

Time Needed for Lesson: 90 minutes

Background Knowledge Needed: Some understanding of Geometry, and Geometer’s Sketchpad

Directions: This lesson can be done one long block setting (80 – 90 minutes) or two shorter periods of 45 – 55 minutes. If done in two shorter periods, you may want to have the “vertical lines” handout ready as they could work on and potential complete that assignment on the second day as well.

Day 1 (Short Class Period) As students enter the room assign them a computer and have them login. Once all of the students are logged in, show them how to access Geometer’s Sketchpad. Review the following tools: Selector, , , and Text and their functions. Next review the directions that are found on the handout “Parallel Lines and Transversals.” Most students should complete through question 4 by the end 45 – 55 minute class period. From there, have students save their work and store it to PowerCourse or to the desktop if students will have access to the same computers the following day.

Day 2 (Short Class Period) Have students login and find their file from the previous class. Once you have worked with the class on getting through question 4, the students should be able to answer most of the remaining questions on their own. Once completed, if you want students to turn in an electronic copy of their work they could load it PowerCourse, or just have them turn in the hard copy of their answers.

Extension/Challenge: On “Parallel Lines and Transversals” and “Vertical Lines” worksheets there is an additional page of questions with additional constructions that students can complete if they finish quickly. This will further students’ understanding of each geometric concept. Created by Don Benn 3 December 2, 2011

Name: ______Computer Number: ______Parallel Lines and Transversals

Before we begin, go to “Sketchpad” in the file menu and select “Preferences”. Change the precision setting on Angle from hundredths to units.

Save your document to the desktop using the naming convention: Name of assignment, Period, Last Name. An example would be: “Parallel Lines P1 Benn.”

Step 1 –Using the line tool, create line AB. Use the text tool to name the points on the line A and B.

Step 2 – Using the point tool, create point C not line AB. Use the text tool to name the point C.

Step 3 – Use the selector tool to deselect all objects. Now use the selection tool to click on point C and line AB. Now use the “construct” menu to create a parallel line.

Step 4 - Using the line tool, construct a line that passes through points A and C to form line AC.

Step 5 – Construct points D, E, F, G, and H as shown here.

Created by Don Benn 4 December 2, 2011

Name: ______Computer Number: ______Step 6 – Select three points at a time to measure each of the eight angles formed. Once you have measured each angle, the measurement will be in highlighted in pink. You have to click somewhere on the white background to deselect the measurement. You may now select the next three points to measure the next angle. List the name and measures of each of the angles in the chart below as they appear on your sketch.

Name of the Angle Measure 1 Measure 2 Measure 3 Angle DCF Angle FCE

Step 7 – Use the selector tool to move point B to a new location. Under the column "Measure 2" write down the new measurements of each angle. Repeat this process again moving point B and complete column “Measure 3”.

Question 1 – What angles are Congruent to each other? (write each set of congruent angles on either side of the bar)

Question 2 - Are the angles that are congruent, always congruent?

Question 3 – Since angles FCE and CAB are known as Corresponding Angles, list other sets of corresponding angles.

Question 4 – Explain in your own words what you observe about corresponding angles.

Question 5 – Angles ECA and CAG can be described as a pair of Alternate Interior Angles. List all of the pairs of alternate interior angles in your construction.

Question 6 - Explain in your own words what you observe about alternate interior angles.

Created by Don Benn 5 December 2, 2011

Name: ______Computer Number: ______Question 7 – Angles FCE and HAG are known as Alternate Exterior Angles. What are the other pairs of alternate exterior pairs?

Question 8 – Explain in your own words what you observe about alternate exterior angles.

Question 9 – How many angles must you know in order to find all 8 angles when working with a transversal line that passes through parallel lines?

Question 10 – What do you notice about the sum of angles DCA and GAC?

Question 11 – There is a special name for this angle relationship when working with parallel lines and transversals. What is it called and is it always true? (Hint: It is not called supplemental.)

Question 12 – When are all of the angles congruent to each other?

Question 13 – Based on the answer from question 12, what do we call two lines that intersect at that angle?

Question 14 – If a transversal intersects the first line in a set of parallel lines at 900, will it intersect all the other parallel lines at the same angle?

Question 15 – What happens to the lines, when one angle is formed by the intersection of one of the parallel lines and the transversal is equal to 00?

Created by Don Benn 6 December 2, 2011

Name: ______Computer Number: ______Part 2 – Is The Converse True?

Step 1 – Go to the “File” menu and select “Document Options”. Click on the drop down menu “Add Page” and then select “Blank Page”. The number 2 will appear in the list. In the “Page Name” box change “2” to “Is The Converse True” and also change “1” to “Parallel Lines ”.

Step 2 – Draw two lines that are not parallel and a third line that will transverse the first two lines.

Step 3 – Now add points using the point tool to have your non-parallel lines and transversal to be similar to the drawing to the right. Please add them and name the points as shown below. Remember that to change a point’s name, double click on the point when the text tool is selected and it will allow you to change the name. If you click in the order shown below then the name of the point should match.

Step 4 – Use the selector tool to measure the 8 angles again.

Step 5 – Now move the lines that are not parallel till the angles match the conjectures you measured in the table. Choose one of the following columns from Step 6 from the “Parallel Lines and Transversal” columns Measure 1, Measure 2, or Measure 3 columns.

Step 6 – Lines that have the same slope are parallel to one another. Measure your slope using the selector tool. Select the two lines that are parallel and select “Measure” from the File menu and select “Slope”. When you do this, a coordinate plane will appear along with the measurement for the slope of the lines.

Created by Don Benn 7 December 2, 2011

Name: ______Computer Number: ______Question 1 – Write a conjecture describing when you know that when two slopes are the same so that the previous conjectures from Part 1 to be true.

Question 2 – From the previous investigation Angles ECA and angle BAC are known as consecutive interior angles. What pairs of angles from this investigation are consecutive interior angles?

Question 3 – What is the sum of the angles that form consecutive interior angles?

Question 4 – Write a conjecture describing the relationship of these consecutive interior angles.

Question 5- From the previous investigation Angles FCD and HAG are known as consecutive exterior angles. What pairs of angles from this investigation are consecutive exterior angles?

Question 6 – What is the sum of the angles that form consecutive interior angles.

Question 7 – Write a conjecture describing the relationship of these consecutive exterior angles.

Created by Don Benn 8 December 2, 2011

Name: ______Computer Number: ______Vertical and Linear Pairs

Step 1 – Start a new document. Use the line tool (not the segment tool) to create line AB. Then starting from point A, create a second line AC. If you did this correctly, when you point A, you affect both lines. If you created segments, you have to start over again.

Step 2 – Next name the points by clicking on the name tool and click on the points. You should have point A at the intersection, and points B and C.

Step 3 - Click on the point tool and make a point D on line AB so that point A is between points B and D. Repeat this process with point E so on line AC where point A is between points C and E.

Step 4 – Choose the selector tool and click on Points B, A and C in that order. Now select “Measure” from the file menu and choose “Angle”. When you have done this, click anywhere in the white area. Repeat this process so that you measure angle CAD, angle DAE, and angle EAB.

Question 1 – Does it matter what order you select your points when measuring your angles?

Question 2 – What letter must always be selected second? Why is this important?

Step 5 – Drag points B and C around the page.

Question 3 – What do you notice about the relationship among the angles you measured?

Step 6 – By definition, angle BAC and angle CAD are known as linear pairs. A linear pair is formed by two angles when they form a ______.

Question 4 – List all sets of angles that form a linear pair.

Created by Don Benn 9 December 2, 2011

Name: ______Computer Number: ______

Question 5 – What do you notice about the relationship between angles in a linear pair? (This is known as the “Linear Pair Conjecture”)

Step 7 – Use the “Number” menu to calculate the sum of your list of angles from question 4. To do this, click on “Number” and then select “Calculate”. Now click on one the angles you measured from Step 4 then type in a plus sign and click on another angle that will form a linear pair. Repeat this process till you have calculated all of the linear pairs.

Question 6 – When are the linear pairs equal to each other?

Question 7 – What do you notice about angle BAC and angle DAE?

Question 8 – What do you notice about angle CAD and angle EAB?

Question 9 – This relationship is known as the Vertical Angles. Write a conjecture for this relationship.

Created by Don Benn 10 December 2, 2011

Name: ______Computer Number: ______Extension

Problem 1 – When two lines intersect, they form four angles. If you know one angle’s measure, you can determine the other three angle measures. Suppose you have three lines that now intersect in a single point.

Question 1 - How many angles are now formed?

Question 2 – What is the minimum number of angles do you need to know in order to find all of the known angles?

Question 3 – At what angle, when three lines intersect at one point, create congruent angles?

Problem 2 – Now let’s look at what happens when four lines intersect at the same point.

Question 4 - How many angles are now formed?

Question 5 – What is the minimum number of angles do you need to know in order to find all of the known angles?

Question 6 – What will the angles be congruent on all of angles formed by the intersection of the lines?

Problem 3 – Looking back at your answers for questions 1 – 6 for the extension, let’s see if you can develop a general equation using your results. Fill in the chart below. Number of Lines Number of Angles Formed Number of angles measures needed 2 4 1 3 4 5 N

Created by Don Benn 11 December 2, 2011