Participant Packet Modules Developed by Macomb County ...mlhartma/6thSession3_Geometry.pdf ·...

Mathematics – Grade Level Assessments and Content Expectations

Grade Six - Session 3

Applying Basic Properties of Lines, Angles, and Triangles

Participant Packet

Modules Developed by Macomb County Teachers

under the leadership of Marie Copeland

Developers

M-GLAnCE Project Directors Debbie Ferry Macomb ISD Mathematics Consultant

Carol Nowakowski Retired Mathematics Consultant K-4 Project Coordinator

Marie Copeland Warren Consolidated Macomb MSTC 5-8 Project Coordinator

2004 Project Contributors David Andrews Chippewa Valley Schools

William Ashton Fraser Public Schools

Lynn Bieszki Chippewa Valley Schools

Sharon Chriss Romeo Schools

Kimberly DeShon Anchor Bay School District

Barbara Diliegghio Retired, Math Consultant

Kimberly Dolan Anchor Bay School District

Jodi Giraud L’Anse Creuse Schools

Julie Hessell Romeo Schools

Amy Holloway Clintondale Schools

Barbara Lipinski Anchor Bay School District

Linda Mayle Romeo Schools

Therese Miekstyn Chippewa Valley Schools

James Navetta Chippewa Valley Schools

Gene Ogden Anchor Bay School District

Rebecca Phillion Richmond Comm. Schools

Charlene Pitrucelle Anchor Bay School District

Shirley Starman Van Dyke Public Schools

Ronald Studley Anchor Bay School District

2005 and 2006 Session/Module Developers Carol Nowakowski Retired, Math Consultant

Deb Barnett Lake Shore Public Schools

Luann Murray Genesee ISD

Kathy Albrecht Retired, Math Consultant

Jo-Anne Schimmelpfenneg Retired, Math Consultant

Marie Copeland Warren Consolidated

Terri Faitel Trenton Public Schools

Debbie Ferry Macomb ISD

Mathematics – Grade Level Assessments and Content Expectations

M-GLAnCE – 6th Grade – Applying Basic Properties of Lines, Angles, and Triangles – Participant Packet 1

Instructional Sequence:

Grade 6 – Session #3 Applying Basic Properties of Lines, Angles, and Triangles G.GS.06.01 G.SR.06.05

Understand and apply basic properties of lines, angles, and triangles, including: Triangle inequality Relationships of vertical angles, complementary angles, supplementary angles Congruence of corresponding and alternate interior angles when parallel lines are cut by a transversal, and that such congruencies imply parallel lines Locate interior and exterior angles of any triangle, and use the property that an exterior angle of a triangle is equal to the sum of the remote (opposite) interior angles Know that the sum of the exterior angles of a convex polygon is 360 degrees

Use paper folding to perform basic geometric constructions of perpendicular lines, midpoints of line segments and angle bisectors; justify informally.

Develop basic terminology of geometry

Develop angle relationships (eg. linear pair, adjacent, etc.)

Sum of the exterior angles of a convex polygon is 360 degrees

Sum of the measures of the angles of a triangle

Develop intersecting, perpendicular, and parallel lines

Relationship between interior and exterior angles of a

Triangle Inequality

Paper folding constructions

Parallel lines conjecture


Important Tips

• Have student’s measure angles with a protractor as often as possible. • To help students understand which numbers to read on a protractor, make the relationship to

right, acute, and obtuse angles. • When students are given examples of intersecting lines, make sure that they are exposed to

situations where the intersecting part is not visible. In other words, the lines have to be extended to see the intersecting portion.

• Have the students use some sort of graphic organizer to help them with all the terms they are learning.

• Vary the experiences students have by sometimes using rulers and protractors, patty paper, and the Geometer’s Sketchpad. Keep in mind, even if the students are using the Geometer’s Sketchpad, which can perform the measurements, they can still be required to print a copy of their work and measure the figures with a ruler and protractor.

• When students are exploring the sum of the angles of a triangle, ask them questions such as, if you have an obtuse triangle, give me a description of possible angle measures of the three angles.

• When investigating the parallel line conjecture, do not always present the situation in the typical fashion. Example:

• Sometimes when students are using manipulatives with investigating the triangle inequality, they

try to bend the materials to force a triangle. • Corresponding angles cause difficulties for students. Relate corresponding angles to

translations. Common Misconceptions: • When students extend the sides of polygons to investigate exterior angles, they make inaccurate

extensions. • When investigating triangles inequality situations, students get confused when it is an equality

situation which forms a straight line. • Students think that if their angles in a triangle add up to 177 degrees, it is close enough to 180

degrees. They do not realize that somewhere they made a mistake in measuring and they have to remeasure the angles.

• Students confuse the vocabulary. • Students think if the rays of an angle extend further than another angle, that is what makes the

angle measure larger • Students do not realize that an angle of a triangle extends beyond the triangle


Grade 6: Participant – Applying Basic Properties of Lines, Angles, and Triangles Name of Activity Description of Activity Materials/Transparencies/

Handouts Key Tips for

The Teacher Vocabulary Building

• Students will use Twizzlers and dot paper to begin the development of the vocabulary. For example, they will use Twizzlers to build a convex and non convex polygon, etc. They will also make graphic organizers to benefit their understanding.

• Twizzlers • Colored Pencils • Magnet Tape • Dot Paper • Scissors • Rulers • Lab Packet pp 5 – 11

• Allow some time for students to work with a partner or group.

• Have magnet tape available so students can put their finished products on the board for group discussion.

Sum of the angles of a triangle

• Students will make triangles and rip off the angles to discover the sum of the angles of a triangle. Connections to exterior angles will also be made.

• 2 pieces of paper • Rulers • Scissors • Colored Pencils • Lab Packet pp 12 - 14

• Have students make assorted triangles that are large so they will be easier to work with but still allows for an extension of the exterior angle. They will also duplicate their triangle.

• Go around the room holding up the various triangles to bring out the terms such as obtuse triangle, acute triangle and right triangle.

Patty paper investigations of special angles

• Students will investigate linear pairs, adjacent angles, and the angles formed by 2 parallel lines cut by a transversal

• Lab packet pp 15– 20 • Patty paper • Rulers • Mechanical pencils

• When students cut their parallel lines by a transversal, make sure to tell them not to make the transversal perpendicular.

• When tracing angles, no free hand tracing. Use rulers.

Euclidean Constructions

• Students will perform the 7 basic Euclidean constructions

• Lab packet pp 21 – 24 • Rulers

• Keep reinforcing that even though some of these


Grade 6: Participant – Applying Basic Properties of Lines, Angles, and Triangles Name of Activity Description of Activity Materials/Transparencies/

Handouts Key Tips for

The Teacher with paper folding. • Patty paper

• Mechanical pencils constructions seem too simple, the reason is because paper folding is being used to introduce Euclidean constructions that only use a straightedge and compass.

Triangle Inequality

• Students test different lengths of line segments to determine which ones will form a triangle and why some work and others do not work.

• Commercial products for this activity work best however, straws or line segments made out of card stock are also suitable

• Fasteners • Lab packet pp 25 – 33

• Make sure students build the triangles but be careful so they do not “force” a triangle, especially when the measurements are close.

Sum of exterior angles of a polygon

• Students will use assorted regular polygons and measure interior, exterior, and central angles to investigate the relationships. Also investigate symmetry in the regular polygons.

• Lab packet pp 34 - 40 • Protractors • Rulers • Miras

• The majority of the work in the investigation portion will be spent with regular polygons, however, also discuss non-regular polygons and non-convex polygons.


Basic Ideas of Geometry 1. Points are labeled with capital letters

2. A line segment shows a part of a line with two distinct end points: written EF or FE

3. Since a line extends to infinity in each direction, a line has arrows going in opposite directions to symbolize the extension. Lines are labeled in two different ways. I. One method of labeling a line is to use a small letter. It is written as line j II. Another way to name a line is to use any two points that fall on the line.

This line is written either GH or HG 4. A ray is a part of angle. A ray contains an endpoint at one end and an arrow at the other end. The arrow symbolizes that the ray extends to infinity. It does not indicate the direction the arrow is going. The ray contains not only an endpoint, but also a point somewhere on the ray. The ray below is called ray QR and is written, QR . Notice that the endpoint always comes first when naming a ray.

5. An angle contains two rays with a common endpoint called the vertex.

B

C

D A

E F

j

HG

R Q

L

K

M

Point A, Point B, Point C, Point D

The angle is written KLM∠ or MLK∠ Notice since L is the vertex, it is written in the middle


6. An angle that measures 90 degrees is a right angle NLM∠ is a right angle

N

L M

In a drawing, a right angle is indicated with a square 7. Since NLM∠ is a right angle

the rays LN and LM are perpendicular The symbol for perpendicular is ⊥ So we can say LN ⊥ LM (ray LN is perpendicular to ray LM) 8. An angle which measures between 0º and 90º is an acute angle.

9. An angle which measures between 90º and 180º is an obtuse angle.

N

90 °

L M

N

90 °

L M


10. When you compare the sizes (lengths or areas) of geometric figures the measures are equal or not equal.

11. When you compare the shapes of geometric figures they are congruent or not congruent. (congruent is to have the same measure) Symbol (≅ )

D

F

G

E

D

F

G

E

A B C

A B C

The measure of ∠DEG is equal to the measure of ∠GEF (m∠DEG = m∠GEF)

The length of segment AB is equal to the length of segment BC (AB = BC)

∠DEG is congruent to ∠GEF (∠DEG ≅ ∠GEF)

Line segment AB is congruent line segment BC ( AB ≅ BC )


12. Bisect is to divide into to equal parts

BD bisects AC because AB = BC

EG bisects DEF∠ because ∠DEG ≅ ∠GEF 13. Parallel lines are lines on the same surface that never intersect. The symbol for parallel is OQ PR (line OQ is parallel to line PR)

A B C

D

O Q

RP

D

F

G

E


14. Two angles who share a common vertex and a common ray but no common interior points are called adjacent angles.

ADB∠ and BDC∠ are adjacent angles 3∠ and 4∠ are adjacent angles 15. Two adjacent angles whose sum is 180 degrees form a linear pair.

ADB∠ and BDC∠ form a linear pair 16. Two non adjacent angles formed by intersecting lines are vertical angles.

1∠ and 2∠ are vertical angles 3∠ and 4∠ are vertical angles

A D C

B

A D C

B

3 4

1 2 3

4


17. Two angles whose measures add up to 90 degrees are complementary angles.

1∠ and 2∠ are complementary angles A∠ and B∠ are complementary angles

because their sum is 90º because their sum is 90º 18. Two angles whose measures add up to 180 degrees are supplementary angles

ADB∠ and BDC∠ are supplementary angles R∠ and S∠ are supplementary angles because their sum is 180º because their sum is 180º 19. Two polygons are congruent if they have all of their corresponding angles and sides congruent. The order of the letters in the statement of congruence indicates which segments and angles are corresponding and congruent.

If ΔABC ≅ ΔFED Then: ∠A ≅ ∠F AB ≅ FE ∠B ≅ ∠E BC ≅ ED ∠C ≅ ∠D AC ≅ FD

C D

EF B A

1 2

30°

A B

60°

150 ° A D

30 ° C

B

80°R

100 ° S


20. An exterior angle of a triangle forms a linear pair with an interior angle of a triangle

When CDB∠ is an exterior angle of ACDΔ When ADF∠ is an exterior angle of ACDΔ

C∠ and A∠ are the remote interior angles C∠ and A∠ are the remote interior angles

When GAD∠ is an exterior angle of ACDΔ When HAC∠ is an exterior angle of ACDΔ

C∠ and D∠ are the remote interior angles C∠ and D∠ are the remote interior angles

When KCA∠ is an exterior angle of ACDΔ When MCD∠ is an exterior angle of ACDΔ

A∠ and D∠ are the remote interior angles A∠ and D∠ are the remote interior angles

D

C

A

F

B

C

A

D

A

G

D C D

A

C

H

C K D

A

C

M

D

A


Lab – Sum of the Measures of the Angles of a Triangle 1 – Distribute paper, rulers and scissors. 2 – Have everyone make two large congruent triangles of all sizes and shapes on the paper using a ruler. 3 – Have the students shade in the three interiors of the interior angles using the same three colors for both triangles

4 – Have the students cut one of the triangles out 5 – Go around the room holding up the various triangles so everyone can see the variety of shapes and sizes 6 – Have the students rip off the shaded regions

7. Have the students place the three regions as adjacent angles along a ruler

8. Ask students: What is the sum of the measures of the three angles?


9. Take the second triangle and extend one side to form a linear pair. An exterior angle forms a linear pair with an interior angle

10. How is the exterior angle related to the other two angles of the triangle?

11. Repeat steps 9 and 10 for the other two sides.


Exterior Angles in Triangles

In any triangle, each exterior angle, is equal to the sum of the two nonadjacent interior angles.

1) a + b + c = 180 degrees 1) The sum of the 3 interior angles of any triangle is 180 degrees 2) a + d = 180 degrees 2) Angles that form linear pairs are supplementary, therefore their sum is 180 degrees 3) a + b + c = a + d 3) Substitution of equal quantities (Steps 1 and 2) 4) -a = -a 4) Reflexive 5) b + c = d 5) Addition of equalities (Steps 3 and 4)

b c

a d

Let a, b, and c represent the measures of the three interior angles of any triangle. Let d represent the measure of an exterior angle that forms a linear pair with the interior angle of measure a. Prove: d = b + c


Looking at Special Angles using Patty Paper I. Vertical Angles Definition: The pairs of opposite angles formed by two intersection lines are called vertical angles. For example, in the diagram, ∠ 1 and ∠ 3 are a pair of vertical angles, and ∠ 2 and ∠ 4 are a pair of vertical angles.

Investigation:

Step 1: Fold a line segment on a patty paper. Unfold and fold a second line segment intersecting the first line segment

Step 2: Label the angles as in the diagram above Step 3: Place a second patty paper over the first and copy one angle of a pair of vertical angles. Rotate the copy to see how well it fits over the second angle of the vertical angle pair. Repeat with the other vertical angle pair.

Write a conjecture about the relationship between a pair of vertical angles.


II. Adjacent Angles and Linear Pairs Definition: Two angles are called adjacent angles if they share a common vertex and a common side but no common interior pints. In the diagram ∠ABD is adjacent to ∠DBC

Definition: A pair of adjacent angles formed by two intersecting lines is called a linear pair. For example ∠ 1 and ∠ 2 are a linear pair of angles and ∠ 3 and ∠ 4 are a linear pair of angles.

Investigation:

Step 1: Fold a line segment on a patty paper. Unfold and fold a second line segment intersecting the first line segment

Step 2: Label the angles as in the diagram above Step 3: Use your protractor to find the sum of the angles of each linear pair of angles. How many pairs are there? Write a conjecture about the relationship between a pair of vertical angles.

A D

C

B


III. Parallel Lines Investigation of Angles Step 1: Construct a pair of parallel lines and use a ruler and straight edge to highlight them

Step 2: Draw a third line called a transversal that intersects the parallel lines and is not perpendicular to the parallel lines and label the angles as shown.

Step 3: Use a second piece of patty paper and trace ∠ 1 with a ruler. What angles have the same measure? Step 4: What angle is congruent with ∠ 1 after a translation of ∠ 1 ? Step 5: What angle is congruent with ∠ 1 after a translation of ∠ 1 and a 180 degree rotation of ∠ 1 ? Step 6: Using the second piece of patty paper, trace ∠ 3 with a ruler. What angles have the same measure? Step 7: What angle is congruent with ∠ 3 after a translation of ∠ 3 ? Step 8: What angle is congruent with ∠ 3 after a translation of ∠ 3 and a 180 degree rotation of ∠ 3 ?


Write a definition for the following: A. alternate interior angles B. alternate exterior angles C. corresponding angles. Identify as many as are pictured above

Name of Angles Numbered Pairs of Angles


Finding missing Angles

1.

2.

3.

°

° °

°

m∠ 1 = _______ m∠ 2 = _______ m∠ 3 = _______ m∠ 4 = _______

m∠ 1 = _______ m∠ 2 = _______ m∠ 3 = _______ m∠ 4 = _______ m∠ 5 = _______

m∠ 1 = _______ m∠ 2 = _______ m∠ 3 = _______ m∠ 4 = _______


4.

8 3

4

1 135 ° 9 10

11 12132

146

57

m∠ 1 = _______ m∠ 8 = _______ m∠ 2 = _______ m∠ 9 = _______ m∠ 3 = _______ m∠ 10 = _______ m∠ 4 = _______ m∠ 11 = _______ m∠ 5 = _______ m∠ 12 = _______ m∠ 6 = _______ m∠ 13 = _______ m∠ 7 = _______ m∠ 14 = _______


Euclidean Constructions by Paper Folding Below are the seven basic constructions used to make geometric drawings. Euclid was the first to organize geometry into a logical structure in his book the Elements. 1. To duplicate a given segment 2. To duplicate a given angle

→

→

3. To construct the bisector or an angle 4. To construct a perpendicular from a point to a line

5. To construct a perpendicular through 6. To construct a perpendicular a point on a line bisector of a given segment

7. To construct a line through a given point parallel to a given line.


Each of the seven constructions using patty paper. 1. To duplicate a given segment Trace the segment onto a new piece of patty paper.

2. To duplicate a given angle Trace the angle onto a new piece of patty paper

3. To construct the bisector or an angle I. Use your straightedge to draw an angle on a piece of patty paper

II. Fold one of the rays of the angle onto the other ray starting at the vertex

III. Open up the paper and make the ray from the vertex along the crease darker by tracing it with a ruler and pencil.


4. To construct a perpendicular from a point to a line I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil to draw in the line segment. Put a point on the patty paper that is not on your line segment.

II. Fold the line segment on top of itself so that the fold contains the given point. Unfold the paper and draw a line through the crease with a ruler and pencil.

5. To construct a perpendicular through a point on a line I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil to draw in the line segment. Put a point on the patty paper that is on your line segment.



6. To construct a perpendicular bisector of a given segment I. Use a straightedge to draw a line segment on a piece of patty paper and put the endpoints on the ends as shown below.

II. Fold the paper so that one endpoint lies on top of the other end point. Unfold the paper and draw in the line segment

7. To construct a line through a given point parallel to a given line. I. Fold a line segment on a patty paper from one side of the paper to the other. Use a ruler and pencil to draw in the line segment. Put a point on the patty paper that is not on your line segment.


III. Fold the new line onto itself through the given point. Unfold the paper and draw a line through the crease with a ruler and pencil.


Are These Triangles? Group #1 Given three possible measurements for the side of a triangle, use the bars and connectors to find out if they do make a triangle.

Case Length(cm) Length(cm) Length(cm) Is it a Triangle?

1

31

22

11

2

15.5

7.75

20

3

31

11

13.5

4

20

11

27

5

7.75

17

27

6

17

13.5

27

7

31

15.5

17

8

31

7.75

20

9

15.5

11

13.5

10

7.75

29

13.5

11

22

20

17

12

31

13.5

27

What conjecture can you make about the relationship between the sides of a triangle that would guarantee you that your three lengths always made a triangle?




1

22

20

11

2

31

17

13.5

3

15.5

22

7.75

4

31

20

11

5

20

13.5

27

6

7.75

17

13.5

7

15.5

20

27

8

31

15.5

22

9

7.75

13.5

27

10

31

20

13.5

11

22

17

27

12

15.5

22

13.5





1

15.5

20

11

2

22

17

11

3

31

22

7.75

4

31

22

27

5

20

17

11

6

31

7.75

13.5

7

15.5

22

17

8

7.75

11

13.5

9

22

7.75

11

10

31

17

27

11

17

11

27

12

22

7.75

20





1

31

15.5

27

2

22

11

13.5

3

7.75

20

17

4

11

13.5

27

5

31

11

27

6

15.5

7.75

13.5

7

22

7.75

13.5

8

20

11

13.5

9

15.5

20

13.5

10

31

17

11

11

31

15.5

7.75

12

22

11

27





1

22

20

13.5

2

31

22

20

3

20

17

27

4

31

20

17

5

22

13.5

27

6

15.5

7.75

17

7

31

15.5

11

8

15.5

7.75

17

9

15.5

13.5

27

10

7.75

20

27

11

20

17

13.5

12

22

7.75

13.5





1

7.75

11

27

2

15.5

20

17

3

31

15.5

20

4

15.5

22

11

5

31

7.75

11

6

15.5

22

20

7

22

20

27

8

17

11

13.5

9

7.75

17

11

10

22

17

13.5

11

15.5

17

27

12

15.5

11

27





1

31

7.75

27

2

31

15.5

13.5

3

15.5

7.75

13.5

4

15.5

17

11

5

31

22

17

6

31

7.75

17

7

15.5

17

13.5

8

7.75

20

11

9

31

20

27

10

22

7.75

17

11

15.5

22

27

12

31

22

13.5



Use the triangle conjecture to mathematically verify which lengths of sides, from the combinations in your group, form triangles and which ones do not form triangles for the first five cases. Case #1 Case #2 Case #3 Case #4 Case #5


Number of Sides and Name of Regular Polygon

Measure of one Central

Angle

Measure of one Interior

Angle

Sum of all of the Interior

Angles

Measure of One Exterior

Angle

Sum of all of the Exterior

Angles – One at each

Vertex


Measure and complete the chart


PARALLEL LINE CONJECTURE

1.) Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees and the precision for angles is set to units.

2.) Use the line tool and make a horizontal line towards the bottom of the screen. 3.) Use the selection arrow tool and click anywhere on the screen to deselect. 4.) Use the point tool and make a point above the horizontal line. 5.) Use the selection arrow tool and highlight the line. The point above the line should already be

highlighted. If the point is not highlighted, then click on it. 6.) Go to the Construct Menu and select Parallel Line. Use the selection arrow tool and click to

deselect. 7.) Use the line tool and make a transversal through the parallel lines. 8.) Use the selection arrow tool and click to deselect. 9.) Use the selection arrow tool and highlight the point on the parallel line. Go to the Display

Menu and select Hide Point. 10.) Use the selection arrow tool and highlight one of the parallel lines and the transversal. Go to

the Construct Menu and select Intersection. Repeat with the other parallel line and the transversal.

11,) Use the selection arrow tool and measure all eight angles. Line up together the angles that

are congruent to each other. 12.) Use the selection arrow tool and highlight all eight angle measurements. Go to the Graph

Menu and select Tabulate. A small table appears with columns for all eight angle measurements.

13.) Double click on the table and another row of the table will appear. 14.) Use the selection arrow tool and move the transversal around to create different angle

measurements. 15.) Repeat steps 13 and 14 until you have 10 different angle measurements. 16.) Go to the File Menu and select Print Preview. Make sure your parallel lines cut by the

transversal and table will print on one page. If not, highlight the entire figure and move it underneath the table.

17.) Print two copies of your figure and table measurements.


NAME:_________________________________________ 1.) Look at the angles formed by the parallel lines cut by the transversal you made on the

Geometer’s Sketchpad. Cut out and tape the picture of the parallel lines cut by the transversal that is on your Geometer’s Sketchpad screen.

2.) In each of the nine additional situations of parallel lines cut by the transversal, what do you

notice about the angle measurements? ________________________________________________________________________ 3.) State the Parallel Line Conjecture. ________________________________________________________________________ ________________________________________________________________________ 4.) State the Converse of the Parallel Line Conjecture. ________________________________________________________________________ ________________________________________________________________________

NAMES OF ANGLES PAIRS OF ANGLES


THE SUM OF THE ANGLES OF A TRIANGLE

1.) Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees

and the precision for angles is set to units. 2.) Use the line segment tool and make a triangle. 3.) Use the selection arrow tool and click anywhere on the screen to deselect. 4.) Use the text tool and label the triangle A, B, and C by clicking on the points of the triangle. 5.) Use the selection arrow tool and select points BAC in order to measure angle A. Next, go to

the Measure Menu and select Angle. 6.) Click anywhere on the screen to deselect. 7.) Use the selection arrow tool and select points ABC in order to measure angle B. Next, go to

the Measure Menu and select Angle. 8.) Click anywhere on the screen to deselect. 9.) Use the selection arrow tool and select points BCA in order to measure angle C. Next, go to

the Measure Menu and select Angle. 10.) Click anywhere on the screen to deselect. 11.) Go to the Measure Menu and select Calculate. 12.) Click on the measurement for angle A, click on the addition sign, click on the measurement for

angle B, click on the addition sign, click on the measurement for angle C, and click OK. 13.) Use the selection arrow tool and select all three angle measurements and the sum of all three

angle measurements. (Four items should be highlighted.) 14.) Go to the Graph Menu and select Tabulate. A small table will appear on your screen with

columns for the three angle measurements and the sum of the angles of the triangle. 15.) Double click on the table and another row of the table will appear. 16.) Use the selection arrow tool and move the vertices around to create a different triangle. 17.) Repeat steps 15 and 16 until you have 10 different triangle measurements. 18.) Go to the File Menu and select Print Preview. Make sure your triangle and table will print on

one page. If not, highlight the entire triangle and move it underneath the table. 19.) Print a copy of your triangle and table measurements.


NAME: ___________________________________ 1.) In your original triangle you made on the Geometer’s Sketchpad, what is the sum of the

measurements of the three angles? ________________________________________________________________________ 2.) In each of the nine triangles you created, what is the sum of the measurements of the three

angles? ________________________________________________________________________ 3.) State a generalization of the sum of the angles of all triangles. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________


ADJACENT ANGLES AND LINEAR PAIRS

1.) Go to the Edit Menu and select Preferences. Make sure the unit for angles is set to degrees and the precision for angles is set to units.

2.) Use the line segment tool and make a line segment. 3.) Use the text tool and label the endpoints, Point C and Point D. If necessary, double click on

the label the computer gave the points and rename them. 4.) Use the line segment tool and highlight the line segment CD. Go to the Construct Menu and

select Point On Segment. 5.) Use the test tool and label the point B. If necessary, double click on the label the computer

gave the point and rename it. 6.) Use the line segment tool and make a line segment from point B. 7.) Use the text tool and label the endpoint, Point A. If necessary, double click on the label the

computer gave the point and rename it. 8.) Use the selection arrow tool and click on the points, A, B, and C. Go to the Measure Menu

and select Angle. 9.) Use the selection arrow tool and click on the points, A, B, and D. Go to the Measure Menu

and select Angle. 10.) Use the selection arrow tool and go to the Measure Menu and select Calculate. Click on one

of the angle measurements, click on the addition sign, click on the other angle measurement, and click OK.

11.) Use the selection arrow tool and highlight the two angle measurements and the sum of the

angle measurements. Go to the Graph Menu and select Tabulate. A small table will appear on your screen with columns for the two angle measurements and the sum of the two angles.

12.) Double click on the table and another row of the table will appear. 13.) Use the selection arrow tool and move the endpoints around to create different angle

measurements. 14.) Repeat steps 12 and 13 until you have 10 different angle measurements. 15.) Go to the File Menu and select Print Preview. Make sure your linear pair and table will print on

one page. If not, highlight the entire figure and move it underneath the table. 16.) Print a copy of your figure and table measurements. 17.) Go to the File Menu and select New Sketch. Use the line segment tool and make two line

segments that intersect.


18.) Use the selection arrow tool and click anywhere on the screen to deselect. 19.) Use the selection arrow tool and highlight both line segments, go to the Construct Menu and

select Intersection. 20.) Use the text tool and label the intersection point E. Label one line segment AB and the other

segment CD. 21.) Use the selection arrow tool and select points BED. Next, go to the Measure Menu and select

Angle. 22.) Click anywhere on the screen to deselect. 23.) Use the selection arrow tool and select points AEC. Next, go to the Measure Menu and select

Angle. 24.) Click anywhere on the screen to deselect. 25.) Use the selection arrow tool and select points AED. Next, go to the Measure Menu and select

Angle. 26.) Click anywhere on the screen to deselect. 27.) Use the selection arrow tool and select points CEB. Next, go to the Measure Menu and select

Angle. 28.) Click anywhere on the screen to deselect. 29.) Identify each linear pair of angles. 30.) Use the selection arrow tool and go to the Measure Menu and select Calculate. Focus on one

linear pair at a time. Click on one of the angles in the linear pair, click on the addition sign, click on the other angle, click OK.

31.) Repeat step 30 with the other linear pair. 32.) What is the sum of each of the linear pairs? 33.) Use the selection arrow tool and select both angles of one linear pair and their sum, then

select both angles of the other linear pair and their sum. 34.) Go to the Graph Menu and select Tabulate. A small table will appear on your screen with

columns for the measurements of each angle of the linear pair and their sum. 35.) Double click on the table and another row of the table will appear. 36.) Use the selection arrow tool and move the endpoints around to create different angle

measurements.


37.) Repeat steps 35 and 36 until you have 10 different angle measurements. 38.) Go to the File Menu and select Print Preview. Make sure your intersecting lines and table will

print on one page. If not, highlight the entire figure and move it underneath the table. 39.) Print a copy of your figure and table measurements.


NAME: _________________________________________ 1.) Look at the angles formed by the intersecting lines you made on the Geometer’s Sketchpad.

Identify and name each of the linear pairs of angles. ________________________________________________________________________ 2.) What do you notice about the sum of each linear pair of angles? ________________________________________________________________________ 3.) In each of the nine additional situations of intersecting lines, what do you notice about the sum

of each linear pair of angles? ________________________________________________________________________ 4.) State the conjecture about the sum of the measures of a linear pair of angles. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________


INTERSECTING LINES AND VERTICAL ANGLES

1.) Go to the Edit Menu and select Preferences. Make sure the units for angle is set to degrees and the precision for angles is set to units.

2.) Use the line segment tool and make two line segments that intersect. 3.) Use the selection arrow tool and click anywhere on the screen to deselect. 4.) Use the selection arrow tool and highlight both line segments, go to the Construct Menu and

select Intersection. 5.) Use the text tool and label the intersection point E. Label one line segment AB and the other

segment CD. 6.) Use the selection arrow tool and select points BED. Next, go to the Measure Menu and select

Angle. 7.) Click anywhere on the screen to deselect. 8.) Use the selection arrow tool and select points AEC. Next, go to the Measure Menu and select

Angle. 9.) Click anywhere on the screen to deselect. 10.) Use the selection arrow tool and select points AED. Next, go to the Measure Menu and select

Angle. 11.) Click anywhere on the screen to deselect. 12.) Use the selection arrow tool and select points CEB. Next, go to the Measure Menu and select

Angle. 13.) Click anywhere on the screen to deselect. 14.) Use the selection arrow tool and select all four angle measurements. 15.) Go to the Graph Menu and select Tabulate. A small table will appear on your screen with

columns for the four angle measurements. 16.) Double click on the table and another row of the table will appear. 17.) Use the selection arrow tool and move the endpoints around to create different angle

measurements. 18.) Repeat steps 16 and 17 until you have 10 different angle measurements. 19.) Go to the File Menu and select Print Preview. Make sure your intersecting lines and table will

print on one page. If not, highlight the entire figure and move it underneath the table. 20.) Print a copy of your figure and table measurements.


NAME: _________________________________________ 1.) Look at the angles formed by the intersecting lines you made on the Geometer’s Sketchpad.

Identify and name the pairs of vertical angles. ________________________________________________________________________ 2.) What do you notice abut each pair of vertical angles? ________________________________________________________________________ 3.) In each of the nine additional situations of intersecting lines, what do you notice about each

pair of vertical angles? ________________________________________________________________________ 4.) State the Vertical Angle Conjecture. ________________________________________________________________________ ________________________________________________________________________ ________________________________________________________________________

Participant Packet Modules Developed by Macomb County ...mlhartma/6thSession3_Geometry.pdf ·...

Documents

Transcript of Participant Packet Modules Developed by Macomb County ...mlhartma/6thSession3_Geometry.pdf ·...