Name Period Date - bloomath.wikispaces.com · LESSON 1.2 Investigation • Virtual Pool ... You will need: the worksheet Poolroom Math, a protractor Pocket billiards, or pool, is

Discovering Geometry Investigation Worksheets

©2015 Michael Serra

LESSON 1.1 1

Investigation • Mathematical Models

Name ____________________________________________ Period ____________ Date __________________

In this lesson, you encountered many new geometry terms. In this investigation you will work as a group to identify models from the real world that represent these terms and to identify how they are represented in diagrams.

Step 1 Look around your classroom and identify examples of each of these terms: point, line, plane, line segment, congruent segments, midpoint of a segment, and ray.

Step 2 Identify examples of these terms in the photograph on page 28 in your textbook.

Step 3 Identify examples of these terms in the figure below.

Step 4 Explain in your own words what each of these terms means.

T E A

M

Discovering Geometry Investigation Worksheets ©2015 Michael Serra

2 LESSON 1.2

Investigation • Virtual Pool

Name ____________________________________________ Period ____________ Date __________________

You will need: the worksheet Poolroom Math, a protractor

Pocket billiards, or pool, is a game of angles. When a ball bounces off the pool table’s cushion, its path forms two angles with the edge of the cushion. The incoming angle is formed by the cushion and the path of the ball approaching the cushion.

The outgoing angle is formed by the cushion and the path of the ball leaving the cushion. As it turns out, the measure of the outgoing angle equals the measure of the incoming angle.

Use your protractor to study these shots on the diagram below.

Step 1 Use your protractor to find the measure of /1. Which is the correct outgoing angle? Which point—A or B —will the ball hit?

Step 2 Which point on the cushion—W, X, or Y—should the white ball hit so that the ray of the outgoing angle passes through the center of the 8-ball?

Outgoingangle

Cushion

Incomingangle

1

Q

A

WXY

C P

B


LESSON 1.2 3

Investigation • Virtual Pool (continued)

Step 3 Compare your results with your group members’ results. Does everyone agree?

Step 4 How would you hit the white ball against the cushion so that the ball passes over the same spot on the way back?

Step 5 How would you hit the ball so that it bounces off three different points on the cushions without ever touching cushion CP?


4 LESSON 1.3

Investigation 1 • Defining Angles

Name ____________________________________________ Period ____________ Date __________________

Here are some examples and non-examples of special types of angles.

Step 1 Write a definition for each boldfaced term in the space below each set of diagrams. Make sure your definitions highlight important differences.

Step 2 Trade definitions and test each other’s definitions by looking for counterexamples.

Step 3 If another group member finds a counterexample to one of your definitions, write a better definition. As a group, decide on the best definition for each term.

Step 4 As a class, agree on common definitions. Add these to your notebook. Draw and label a picture to illustrate each definition.

Right AngleRight Angle

Right angles

90°

468

1058

Not right angles

898

338

568

Acute angles Not acute angles

1008

918

Acute Angle


LESSON 1.3 5

Investigation • Defining Angles (continued)

Obtuse Angle

Complementary Angles

m/1 1 m/2 5 90°

Supplementary Angles

m/3 1 m/4 5 180°

3 42

1

Pairs of complementary angles: /1 and /2 /3 and /4

3 4

1

2 408

G

H

528

Not pairs of complementary angles: /G and /H /1 and /2 /3 and /4

3

2

1

4

Pairs of supplementary angles: /1 and /2 /3 and /4

1 2 3 4

5

Not pairs of supplementary angles: /1, /2, and /3 /4 and /5

918

1018

1298

Obtuse angles Not obtuse angles

388

898

m/1 1 m/2 90°

m/4 1 m/5 180°


6 LESSON 1.3

Investigation • Defining Angles (continued)

Vertical Angles

Linear Pair of Angles

34

1 2

AE

D

BC

Pairs of vertical angles: /1 and /2 /3 and /4 /AED and /BEC /AEC and /DEB

1 2

3

4

7 8

5 6

9 10

Not pairs of vertical angles: /1 and /2 /3 and /4 /5 and /6 /7 and /8 /9 and /10

1 2

A D

BC

E 3

4

Linear pairs of angles: /1 and /2 /3 and /4 /AED and /AEC /BED and /DEA

1 2

56

3

308

1508

4 A

B

Not linear pairs of angles: /1 and /2 /3 and /4 /5 and /6 /A and /B


LESSON 1.3 7

Investigation 2 • Creating Angles with Patty Paper

Name ____________________________________________ Period ____________ Date __________________

You will need: patty paper

Step 1 Fold and crease a random line through your patty paper.

Step 2 Open your patty paper and fold another random line.

Step 3 Show the other members of your group

a. an acute angleb. an obtuse anglec. a pair of vertical angles d. a linear pair of anglese. a pair of supplementary angles.

Step 4 On a second patty paper, fold and crease a random line.

Step 5 Fold your first crease on top of itself. What kind of angle does this appear to be? Test your conjecture by placing the corner of another patty paper into the angle formed by the intersection of the two creases. Explain why the angle turned out to have that angle measure. Place a dot at the vertex of the angle.

Step 6 Fold a third crease passing through the point where the two lines intersect (the vertex of your special angle).

Step 7 Show the other members of your group

a. two acute anglesb. two obtuse anglesc. two right anglesd. two different pairs of vertical angles e. a linear pair of anglesf. two different pairs of supplementary anglesg. two different pairs of complementary angles.

Step 1

Step 2

Step 4

Step 5

Step 6


8 LESSON 1.4

Investigation • Special Polygons

Name ____________________________________________ Period ____________ Date __________________

Write a good definition of each boldfaced term. Discuss your definitions with others in your group. Agree on a common set of definitions for your class and add them to your definitions list. In your notebook, draw and label a figure to illustrate each definition.

Equilateral Polygon

Equiangular Polygon

Regular Polygon

Equilateral polygons Not equilateral polygons

Equiangular polygons Not equiangular polygons

Regular polygons Not regular polygons


LESSON 1.5 9

Investigation • Triangles

Name ____________________________________________ Period ____________ Date __________________

Write a good definition of each boldfaced term in the space below each set of diagrams. Discuss your definitions with others in your group. Agree on a common set of definitions for your class and add them to your definitions list. In your notebook, draw and label a figure to illustrate each definition.

Right Triangle

Acute Triangle

Obtuse Triangle

Right triangles

75°

90° 15°

26° 64°

Not right triangles

58°

87° 35°

65°

91°

24°

858

808

558

408 728

288

638

518

668

Acute triangles Not acute triangles

50°

40°

27° 122°

31°

41°

112°

22°

118°

40°

27°

Obtuse triangles Not obtuse triangles

33°

57°

26°72°

104°

116°

68°

66°

88°


10 LESSON 1.5

Investigation • Triangles (continued)

Scalene Triangle

Equilateral Triangle

Isosceles Triangle

Scalene triangles

15

17

28 20

11

8 7

6

8

Not scalene triangles

11

4

8

65

8

8 8

Equilateral triangles Not equilateral triangles

11

11

10

Isosceles triangles

14

14

17

Not isosceles triangles

12

13

5

30

22 23


LESSON 1.5 11

Investigation 2 • Creating Triangles with Patty Paper

Name ____________________________________________ Period ____________ Date __________________

You will need: patty paper

Step 1 Fold and crease a random line through your patty paper.

Step 2 Open your patty paper and fold another random line.

Step 3 Fold another random line to create a triangle. Show the other members of your group your triangle.

a. Is it an acute triangle, obtuse triangle or right triangle? How do you convince the others in your group?

b. Is it a scalene triangle, isosceles triangle or equilateral triangle? How do you convince the others in your group?

Step 4 On a second patty paper, fold and crease a random line.

Step 5 Fold your first crease on top of itself to create a right angle.

Step 6 Fold a third crease to create an isosceles right triangle. Explain to others in your group what you did so that you knew the triangle must be isosceles (“because it looks it” is not a good enough reason!).

Step 1Step 2 Step 3

Step 4Step 5 Step 6


12 LESSON 1.6

Investigation • Special Quadrilaterals

Name ____________________________________________ Period ____________ Date __________________

Write a good definition of each boldfaced term in the space below each set of diagrams. Discuss your definitions with others in your group. Agree on a common set of definitions for your class and add them to your definitions list. In your notebook, draw and label a figure to illustrate each definition.

Trapezoid

Kite

Trapezoids Not trapezoids

Kites

Not kites


LESSON 1.6 13

Investigation • Special Quadrilaterals (continued)

Parallelogram

Rhombus

Parallelograms Not parallelograms

Rhombuses Not rhombuses


14 LESSON 1.6

Investigation • Special Quadrilaterals (continued)

Rectangle

Square

Rectangles

90° 90°

90°90°

Not rectangles

S R

P Q

A

B

C

D

Squares Not squares


LESSON 1.6 15

Investigation 2 • Creating Special Quadrilaterals

Name ____________________________________________ Period ____________ Date __________________

You will need: heavy paper, a straightedge, scissors

Step 1 Draw and cut out two congruent acute scalene triangles on heavy stock paper.

Step 2 Can you arrange them into a parallelogram? Sketch your results, marking all the congruent sides and angles. How do you convince the others in your group that your shape is a parallelogram (“because it looks it” is not a good enough reason!)?

Step 3 Can you arrange them into a kite? Sketch your results, marking all the congruent sides and angles. How do you convince the others in your group that your shape is a kite?

Step 4 Draw and cut out two congruent obtuse isosceles triangles on heavy stock paper.

Step 5 Which special quadrilaterals can you create with these two congruent triangles? Sketch your results, marking all the congruent sides and angles. How do you convince the others in your group that your shape is what you claim it to be?

Step 1

Step 4


16 LESSON 1.7

Investigation • Defining Circle Terms

Name ____________________________________________ Period ____________ Date __________________

Step 1 Write a good definition of each boldfaced term in the space below each set of diagrams. Discuss your definitions with others in your group. Agree on a common set of definitions as a class and add them to your definitions list. In your notebook, draw and label a figure to illustrate each definition.

Chord

Diameter

A

B

D

CE

F

H

G

IJ

Chords:

AB CD EF GH, , , , and IJ

T

UV

W

P

QR

S

Not chords:

PQ RS TU, , , and VW� ��

E

C

A

F

D

B

O

Diameters:

,AB CD, and EF

W

R

S

P VT

U

Q

Not diameters:

PQ RS TU� ��

, , , and VW


LESSON 1.7 17

Investigation • Defining Circle Terms (continued)

Tangent

Note: You can say AB� ��

is a tangent, or you can say AB� ��

is tangent to circle O. The point where the tangent touches the circle is called the point of tangency.

Step 2 Can a chord of a circle also be a diameter of the circle? Can it be a tangent? Explain why or why not.

Step 3 Can two circles be tangent to the same line at the same point? Draw a sketch and explain.

F E

A

B C

D

Tangents:

,AB CD� ��

, and EF��

P Q

T

U S

R

V

W

Not tangents:

, ,PQ RS TU� ��

, and WV� ��


18 LESSON 1.8

Investigation • Space Geometry

Name ____________________________________________ Period ____________ Date __________________

Use the space below each statement for your sketches and answers.

Step 1 Make a sketch or use physical objects to demonstrate each statement in the list below.

Step 2 Work with your group to determine whether each statement is true or false. If the statement is false, draw a picture and explain why it is false.

1. For any two points, there is exactly one line that can be drawn through them.

2. For any line and a point not on the line, there is exactly one plane that can contain them.

3. For any two lines, there is exactly one plane that contains them.


LESSON 1.8 19

Investigation • Space Geometry (continued)

4. If two coplanar lines are both perpendicular to a third line in the same plane, then the two lines are parallel.

5. If two planes do not intersect, then they are parallel.

6. If two lines do not intersect, then they are parallel.

7. If a line is perpendicular to two lines in a plane, and the line is not contained in the plane, then the line is perpendicular to the plane.

Investigation 1 • The Basic Properties of a Reflection

Name ____________________________________________ Period ____________ Date __________________

In this investigation you’ll model reflection with patty paper and discover important properties of this transformation.

Step 1 Place a line of reflection on a piece of patty paper. Draw a polygon next to it with one of the vertices on the line of reflection.

Step 2 Fold your patty paper along the line of reflection and create the reflected image of your polygon by tracing it. Open up the patty paper.

Step 3 Draw segments connecting each vertex with its image point.

Step 1 Step 2 Step 3

Step 4 Use your patty paper investigation to explain to other members of your group what things are the same in both the original figure and its image. Lengths? Angles? Orientation? What is true about the segments connecting points and their images?

Step 5 In your group discuss which of the following statements are true. If true, demonstrate an example to the other group members. If false, sketch a counterexample. Explain to other group members why your counterexample demonstrates that the statement is false.

a. If AB is reflected over line m creating the image A B′ ′ then AB A B′ ′.

b. If /A is reflected over line m creating the image /A′ then /A /A′.

c. If polygon P is reflected over line m creating the image polygon P′ then P P′.

d. A reflection transformation is a rigid transformation or isometry.


20 LESSON 1.9

e. If a set of points are collinear, then their reflected images are also collinear.

f. If point N is between A and B, then the reflected image of N is between the images of A and B.

g. If the image of a point A reflected over line m is B, then the image of B reflected over line m is A. (If A is on m, then the image of A coincides with A.)

h. If the clockwise order of the vertices of a quadrilateral ABCD are A, then B, then C, then D, and back to A, then the clockwise order of points of the reflected image A′B′C′D′ is the same: A′, then B′, then C′, then D′, and back to A′.

i. Every point and its reflected image are always the same distance from the line of reflection. In other words, the line of reflection always bisects any segment connecting a point to its reflected image point.

j. The segment connecting a point to its reflected image is always perpendicular to the line of reflection.

k. The line of reflection bisects every segment connecting a point and its image point.


LESSON 1.9 21


22 LESSON 1.9

Investigation 2 • The Basic Properties of a Translation

Name ____________________________________________ Period ____________ Date __________________

You will need: a straightedge, a compass, patty paper

In this investigation you’ll model translation with patty paper and discover important properties of this transformation.

Step 1 Draw a quadrilateral in one corner of your patty paper. From one of your vertices (label it A), draw a ray to the edge of your patty paper. This will be the direction of the translation. Place a point B on that ray. The distance from A to B is the translation distance. This distance, together with its direction is called the translation vector AB.

Step 2 Place a second patty paper on top of the first and make a copy of the quadrilateral, the ray, and the translation vector. Place your copy beneath the original and, using the translation vector as a guide, translate the second copy, keeping the rays aligned until the point A of the quadrilateral copy is over the point B on the original ray. Copy the image of the bottom quadrilateral onto the top patty paper.

Step 3 Draw segments connecting each vertex with its image point.

Step 4 Use your patty paper investigation to explain to other members of your group what things are the same in both the original figure and its image. Lengths? Angles? Orientation?


a. If PQ is translated creating the image P Q′ ′ then PQ P Q′ ′.

b. If /R is translated creating the image /R′ then /R /R′.

Step 2

B

B

A

Step 1

B

A

Step 1

Step 3

A

B

Step 1


LESSON 1.9 23

c. If polygon S is translated creating the image polygon S′ then polygon S polygon S′.

d. A translation transformation is a rigid transformation or isometry.

e. If a set of points are collinear, then their translated images are also collinear.

f. If point N is between A and B, then the translated image of N is between the images of A and B.

g. If point A is translated by the translation vector PQ creating image B, then the image of B translated by the translation vector QP is A.

h. If the clockwise order of the vertices of a quadrilateral ABCD are A, then B, then C, then D, and back to A, then the clockwise order of points of the translated image A′B′C′D′ is the same: A′, then B′, then C′, then D′, and back to A′.

i. Any point and its translated image point are the same distance apart as any other pair of points and corresponding image points.

j. All the segments connecting a point to its translated image point are parallel.


24 LESSON 1.9

Investigation 3 • The Basic Properties of a Rotation

Name ____________________________________________ Period ____________ Date __________________

You will need: a straightedge, a compass, patty paper

In this third investigation you’ll model rotation with patty paper and discover important properties of this transformation.

Step 1 Draw a quadrilateral in one corner of your patty paper. Place a dot near the center of your patty paper and label it point P. This will be your center of rotation. From point P draw a segment to one of your vertices (label it A).

Step 2 Place a second patty paper on top of the first and make a copy of the quadrilateral, the center of rotation, and the segment PA. Place your copy beneath the original aligning the quadrilaterals and the centers of rotation. Place your pen or pencil on top of the centers of rotation. Rotate the copy until the two quadrilaterals are no longer overlapping. Copy the image of the bottom quadrilateral onto the top patty paper.

Step 3 Draw segments connecting each vertex to the center of rotation P.

Step 4 Use your patty paper investigation to explain to other members of your group what things are the same in both the original figure and its image. Lengths? Angles? Orientation?


a. If PQ is rotated creating the image ′ ′P Q then PQ P Q≅ ′ ′ PQ P Q≅ ′ ′ .

b. If /R is rotated creating the image /R′ then /R /R′.

P

A

Step 1

P

AP

A

Step 2

P

AA

P

Step 3


LESSON 1.9 25

c. If polygon S is rotated creating the image polygon S′ then polygon S polygon S′.

d. A rotation transformation is a rigid transformation or isometry.

e. If three or more points are collinear, then their rotated images are also collinear.

f. If point N is between A and B, then the rotated image of N is between the rotated images of A and B.

g. If point A is rotated clockwise creating image B, then the image of B rotated clockwise is A.

h. If the clockwise order of the vertices of a quadrilateral ABCD are A, then B, then C, then D, and back to A, then the clockwise order of points of the rotated image A′B′C′D′ is the same: A′, then B′, then C′, then D′, and back to A′.

i. Any point and its rotated image point are the same distance apart as any other pair of points and corresponding image points.

j. All the segments connecting a point to its rotated image point are parallel.

Documents

Name Period Date - bloomath.wikispaces.com · LESSON 1.2 Investigation • Virtual Pool ... You will need: the worksheet Poolroom Math, a protractor Pocket billiards, or pool, is