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Foundations of Math II Introduction to Geometry: Vocabulary and Drawing

Conclusions

AcademicsHigh School Mathematics

Mira LabRead and follow the Math II Lab Directions. Remember that you should trace all figures and place all answers onto your own paper. You are working with a partner to share and discuss ideas, but each of you is responsible for writing down your own answers.1) In front of you is a red plastic thing called a Mira. What do you think it is used for?

Why?2) Place the edge of the Mira on the line below. Look through the Mira. What do you see?

3) Trace the figure above onto your paper. Then, looking through the Mira, trace the image of the envelope on the other side of the line.

4) Use the Mira to complete each word or picture (you do not have to retrace these onto your paper). Describe what you see for each. (Hint: Placement of the angled edge of the Mira is very important!)

5) Use the Mira to complete the message to the right. What does it say? 6) Trace the figure below onto your own paper. Draw a reflection of the object

over each of the lines a – d.1

7) Place a mirror in front of you. Write your first name on your paper so that it appears correct when looking in the mirror. Explain how you did it.

8) How could you do the problems in this lab if you did not have a Mira or a mirror?

Mira Lab Part II9) Try out the alternate method from #8 to make a reflection of the figure below.

Remember to traceonto your own paper first!

2

10) Make up a word drawing of your own that can be completed using a Mira. Can you use any letter? Why or why not? Explain the process you used to make up your word drawing.

11) Which of the following figures is a reflection of Figure 1? If it is a reflection, draw the line of reflection. If it is not a reflection, place an X over the figure. Describe your method. (Remember to trace onto your own paper first!)

12) What does a Mira or a mirror have to do with Geometry?

Challenge:The penta-dog below has an interesting tail. Draw a reflection of the penta-dog so that his tail is pointing straight down. Then draw an image so that his tail is pointing directly to the left.

3

1.2 Guided Practice Name _____________________________________________________________

6

What did one tree say to the

other tree?

Vocabulary Words

1)

2)

3)

4)

5)

6)

7)

List statements that describe what you see in the picture above.

J

K

1.2 Practice: Name ______________________________________________________Describing Geometric Drawings

Write ten statements that describe things that you see in the picture above.

1.

2.

3.

4.

5.

6.

7.

8.

9.

10.

7

Gee, I’m a tree!

1.2 Show What You Know! Name ______________________________________________________

Make a sketch of each of the following situations. Be sure to label completely and add dotted lines where appropriate to give a three-dimensional perspective.

1. A vertical line m intersecting a horizontal plane N.

2. A horizontal plane P containing 2 lines, r and s, intersecting in point Q.

3. Three interesting lines l, m and n, intersecting at point X.

4. Two horizontal planes, M and N, with line l intersecting them both.

5. Bring in an example of a point, a line and a plane to class tomorrow.

Algebra Review Solve for x.6. −20=−4 x−6 x

7. 2 (4 x−3 )−8=4+2x

8. 14=−(m−8 )

9. 4 y−2=3 y+1 y

10. −4 (−6 x−3 )=12

8

1.3 Warm Up1. Bill is sitting on a boat on a smooth mountain lake. He sees the scene pictured below, in

which a swan is flying over the lake toward a distant tree. He also sees the swan in the lake. Draw what he sees in the lake. Explain your answer.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii

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1.3 Tissue Box Investigation Name: _____________________________________

Objective: Students will be able to describe in both verbal and pictorial form how lines and planes intersect and do not intersect.I. Give a description for each of the following undefined terms.

1. point:

2. line:

3. plane:

4. space:

II. Define intersection:

III. You will use the following real world items to determine how lines and planes intersect. Please determine which geometric term (from above) could be used to describe each of the items listed below. (For example, a string can represent a line.)1. a piece of paper:2. tissue box:3. pencil: 4. a corner of the tissue box:

5. a corner of piece of paper:6. the edge of a piece paper:7. the edge of a box: 8. a hole in the piece of paper:

IV. Using your real world tools, answer the following questions about intersections. Your goal is to determine how planes and lines can intersect. In the following you will use the materials to help you determine what these intersections look like – a point? a line? a plane? or no intersection at all? 1. Do the top of the tissue box and the bottom intersect? What geometric term describes the

relationship between the top and the bottom of the box?

2. Do the top and the right side of the tissue box intersect? What geometric term describes the intersection of the top and the right side?

3. Using some tape, connect two pieces of paper together like a book. How do the papers intersect? (Use a geometric term to describe the intersection.)

4. What geometric term describes the intersection of three sides of the tissue box?

5. Using tape, connect three sheets of paper like a book. What geometric term describes how they intersect?

6. Can four sides of the tissue box intersect in only one point or line? Explain.

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7. How can a side of the box intersect with an edge of the box? Explain.

8. If you insert the pencil into the paper with the hole punched out, what is the intersection of the pencil and the paper?

9. Does one top edge and one bottom edge of the box intersect? Why or why not?

10.Do the lines labeled AB and CD on your tissue box intersect? What geometric term describes the relationship between these two lines?

V. Thinking back. Use your answers to Part IV to make generalizations about how geometric objects intersect. Describe both in WORDS and with a PICTURE.

1. If two lines intersect, what is the intersection?

2. If two planes intersect, what is the intersection?

3. If three planes intersect, what is the intersection? (Hint: two different answers!)

4. If two lines do not intersect, what do we call them? (Hint: two different answers!)

5. If a plane and a line intersect, what is the intersection? (Hint: two different answers!)

1.3 Practice Name ________________________________________Points, Lines, and Planes

Draw and label a figure for each verbal description.

1. ABand EF intersect plane U at point X, but they do not lie in plane U.

15

A

L

O

V

M

T

H

E

WY

T

2. Lines g and h intersect at point P.

3. Points J, K, and L are in plane H, but point M is not in plane H.

4. Planes P, Q, and R intersect at DE.

Refer to the figure at the right to answer each question.

5. Are points E, V, T, and A coplanar?

6. Name three lines that intersect at point M.

7. What points do plane MATH and ET have in common?

8. Are points L, H, and M collinear?

9. Name a line that is skew to AM .

10. Name a line that is parallel to LH .

1.3 Show What You Know! Name _______________________________________

State whether each object is best described by a point, a line, or a plane.

1. a star in the sky

2. the floor of the gym

3. an overhead power line

4. a pixel on a computer screen

5. the top of your school desk

6. the spine of a book

Use symbols to name each of the following objects in picture to the right.

16

7. a line containing point W

8. two points collinear with point X

9. a line passing through point Y

10. two points coplanar with point Z

11. a point not on plane R.

12. Challenge: two points coplanar with point U

Using colored pencils, label and shade the figure as described.

13. With a yellow pencil, shade a plane. Then label three noncollinear points on

the plane as R, S, and T.

14. With an orange pencil, shade a plane that intersects the yellow plane.

15. What is the intersection of the orange and yellow planes?

16. With a red pencil, label four coplanar points as E, F, G, and H.

17. With a blue pencil, label three collinear points as W, X, and Y.

18. With a green pencil, label four noncoplanar points as L, M, N, O.

Graph and label the following points on the grid below: S(-1, -1), T(0, 4), U(-3, -5), V(2, 5), and W(3, -4). Then answer the questions.

19. Name three noncollinear points.

20. Name three collinear points

21. Name two intersecting lines.

1.4 Guided Practice Examples

Example 1

In the figure below, B is the midpoint of AC. Find x and AC.

A 2x + 12 B 5x +10 C

Example 2

Find x, RS, and ST. R 2x – 4 S 3x +7 T

17

ZV

R

43

Example 3

In the figure below, C is the midpoint of AB. Find x and AC.

5x – 6 2x

A C B

Example 4

If RT = 60, find x, RS, and ST.

3x – 12 2x – 8

R S T

Example 5

In the figure below, G is the midpoint of EF . Find y and EG.

E

3y

G

36 – y

F

18

1.4 Show What You Know! Name _______________________________________

Use the figure below to decide whether each statement is true or false. Explain why each statement is true or false on your own paper!

1.1. PB bisects RS.

2. S is the midpoint of TV .

3. SV TS.

4. W bisects VR.

5. SW is longer than VR.

6. V is the midpoint of TW .

7. WR QV .

8. TB + BW = TW.

Given that point A lies between points C and T, find each missing value. (Hint: Draw and label a diagram for each problem.)9. CA = 6

AT = ?CT = 15

10. CA = 5.2AT = 4.8 CT = ?

11. CA = ?

AT = 512

CT = 1034

12. CA = 22AT = ?CT = 35

For each figure, find x and the indicated segment measure. Show all work on your own paper!13. B is the midpoint of AC.

AB = 4x – 3 BC = xAC = ?

14. E is the midpoint of DF .DF = x + 6DE = x – 1 EF = ?

15. BX bisects TW .WB = x + 5TW = 4x + 5TB = ?

16. DG bisects CT at point O.CO = 2x + 1OT = x + 7CT = ?

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P

T S V

A

Q

B

W

B

R

C

T B WX

D EF

T

D

C

O G

1.5 Warm Up

1. Chris is making an abstract drawing in art class. First, he paints a line as shown. Then, before the paint dries, he folds the paper on the dotted line and unfolds it again.

a. Accurately draw his completed painting.b. What does this problem have to do with Geometry?c. Should Chris expect to receive a scholarship to the NC School of the Arts?

2. Shari buys a rug that has a star on one of the corners. She places it in front of a full-length mirror in her bathroom. The top view is shown below.

d. Draw the reflection of the rug. Explain your method.e. Compare the rug and its reflection. What do you notice?

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii1.5 Guided Practice

Fill in the diagram and find the missing length for each right triangle using the Pythagorean Theorem.

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mirror

1. a = 5, b = 12, c = ?

2. a = 7, c = 25, b = ?

3. b = 15, c = 17, a = ?

4. a = 9, b = 12, c = ?

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1.6 Warm Up

1. Which of the sailboats are images of boat J using a reflection? Explain why or why not for each figure.

2. Jill’s mother asked her to move a heavy metal table. Its top view is shown below, with the positions of its legs marked.

After moving the table, Jill sees to her horror that the lower right leg of the table has made a deep scratch mark on their wood floor as shown below.

Draw the position of the table before and after Jill moved it.


23

1.6 Practice Name _________________________________

Constructing a Tangram

A tangram is a geometric puzzle thought to have been developed in China. The puzzle consists of two large and two small pairs of congruent triangles, a medium-sized triangle, a square, and a parallelogram. The tangram pieces can be used to construct many different shapes. Follow the instructions below to construct your own set of tangrams. Show all calculations on your own paper.

1. Using the graph paper provided, plot and label the points A(0, 8), B(8, 8), C(8, 0), and D (0, 0). Connect the points to form a square.

2. Draw BD.

3. Find the length of BD to the nearest tenth.

4. Find the midpoint of AB and label it E. What are the coordinates of E?

5. Find the midpoint of AD and label it F. What are the coordinates of F?

6. Draw EF .

7. Find the length of EF to the nearest tenth.

8. How does EF compare to BD?

9. Find the midpoint of EF and label it G.

10. Draw CG.

11. Find the length of CGto the nearest tenth.

12. Find the intersection of CG and BD. Label it H.

13. Find HC to the nearest tenth. How does it compare to BD?

14. Find the midpoint of HD and label it J. What are the coordinates of J?

15. Draw FJ .

16. Find the midpoint of HB and label it K. What are the coordinates of K?

17. Draw GK .

18. Cut out the pieces. You should have seven pieces total, each of which is called a tan.

19. Create figures out of your tans. Try to make a boat, a building, an animal, anything else you

can come up with. Choose one figure and trace the outline (outside edges only!) You will

share your outline with a partner tomorrow to see if he/she can recreate it using the

tangrams.

Adapted from “1-6 Constructing a Tangram” in Geometry Teacher’s Activities Kit by Judith A. Muschla and Gary R. Muschla 1.6 Show What You Know! Name ________________________________________

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-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

H IK J K L M N

-4 -3 -2 -1 0 1 2 3 4 5 6 7 8

H I J K L M N

Use the number line to find each measure.1. HK 2. LN 3. LJ 4. KM

Use the Pythagorean Theorem to find the distance between each pair of points.5. 6.

7. K (-2, -6), L(3, 6) 8. S(-3, -1), T(1, 5)

Use the number line to find the coordinate of the midpoint of each segment.9. KM 10. KN 11. HJ 12. JL

Find the coordinates of the midpoint of the segment with the given endpoints.13. B(3, 1), C(5, 5) 14. W(-4, 3), V(5, -3)

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1.7 Warm Up

1. Triangle A’B’C’ is the image of triangle ABC reflected over line m. Draw triangle ABC. Explain your method.

2. Gerry is rearranging the furniture in his living room. He has to leave before he is finished, so he draws the diagram below for his wife to place the end table. Draw the new position of the end table.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii1.7 Show What You Know! Name __________________________________

Find the distance and midpoint between each pair of points.

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m

1) (4, 6), (-2, -1)

2) (-5, 3), (4, -1)

3) (6, 3), (-3, 5)

4) (-3, -4), (2, 0)

Find the coordinates of the missing endpoint of AB given the coordinates of one of the endpoints and the midpoint M.

5) A(3, 6), M(0, 2) 6) B(-1, 5), M(1, 1)

7)

30

1.8 Warm Up

1. Swan d is the pre-image of swan d’. Draw the line of reflection for d to d’. Next, draw the line of reflection that would reverse the reflection (so that swan d’ is the pre-image of swan d). What do you notice?

2. Stand beside your chair. Then move so that you perform a translation. Draw a diagram illustrating you and your motion. Explain why your motion is a translation.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawa

31

1.8 Additional Practice

In the figure, XP and XT are opposite rays. XQ bisects PXS. For each situation, find the value of x and the measure of the indicated angle.

1) mSXT = 4x + 1, mQXS = 2x – 2, mQXT = 125; Find mQXS

2) mPXR = 3x, mRXT = 5x +20; Find mRXT

3) mRXQ = x + 15, mRXS = 5x – 7, mQXS = 3x +5; Find mRXS

4) mTXS = x + 3, mSXR = 2x + 9, mRXP = 42x – 7; Find mPXS

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R

QS

P

X

T

5

1

2

4 7

68 9

1.8 Show What You Know! Name___________________________________

Find the distance and midpoint between each pair of points.

1) (4, 6), (-2, -1)

2) (-5, 3), (4, -1)

3) (6, 3), (-3, 5)

4) (-3, -4), (2, 0)

Name the vertex of each angle.5) 7

6) 6

7) 8

8) 4

Name the sides of each angle.9) 1

10) EFB

11) 5

12) 2

Write another name for each angle.13) 4

14) 7

15) EGC

16)AFE

35

Measure each angle with a protractor and classify as right, acute, obtuse, or straight.

17) YVZ

18) XVW

19) UVW

20) XVZ

21) UVY

Mark the given information on the figure below. Then use the figure to help solve the following problems.

22) If mLQM = 7x – 12 and mNQM = 5x + 10, find x and mLQN.

23) If mOQP = 4x + 16 and mNQO = 6x + 4, find x and mOQP.

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U

Z

Y

X

W

V

L

ON

M

PQ

QM bisects LQN

QO bisects PQN

B

D

E

B

D

E

1.9 Practice Problems

Set 1:

1) Two angles are complementary. If one angle measures 4x + 3 and the other measures 2x + 21, find the measure of each angle.

2) DE is the perpendicular bisector of AC. If AB = 4x + 1 and AC = 3x + 42, find x, AB, and AC.

3) Find the measure of x.

4) DE is the perpendicular bisector of AC. If mDBF = 2x + 6 and mCBF = 2x, find mDBF and mCBF.

37

A C

(3x – 6)27

A C

F

B

D

E

Set 2:

5) Find the measure of x.

6) DE is the perpendicular bisector of AC. If AB = 9x – 7 and AC = 2x + 18, find x, AB, and AC.

7) An angle measures 7x – 13. Its complement measures 4x + 15. Find the measure of each angle.

8)

38

(2x – 1)75

A C

1.10 Warm Up

1. Draw BAC, the image of XYZ when reflected using line p. Then draw segments BX , AY , and CZ. What do you notice?

2. Draw the image of the triangle when translated using arrow a. Explain your method.

Draw the image of the triangle when translated using arrow b. Explain your method.

Draw the image of the triangle when translated using arrow c. Explain your method.

42

a

b

What do you notice?


43

c

1.10 Practice Name___________________________________

Find the missing measures for the numbered angles.

m1 = 53

m2 = ________

m3 = ________

m4 = 28

m5 =

m6 = 44

m7 = ________

m8 = 37

m9 = ________

m10 = ________

m11 = ________

m12 = ________

m13 = ________

m14 = ________

m15 = ________

m16 = ________

m17 = ________

44

1 23

45

6 7

8

9 100 11

AM T

H

S

C

AS bisects HAC

AS MT

12 1314 m< 14 = 2x +6

15

m< 15 = 10x -10

m12 = 4x + 11m13 = 3x + 1

16

17m16 = 5x + 3m17 = 9x – 5

16 and 17 are supplementary

x

z67y

(4x + 50) (2x + 60)

RA

B

T N

1.10 Show What You Know! Name__________________________________________

Find the missing measures. Show all work on a separate piece of paper!

1) 2)

3) 4)

5) 6)

7) 8)

9) Find the measures of two complementary angles A and B if mA = 7x + 4 and mB = 4x + 9.

10) The measure of an angle is five less than 4 times the measure of its supplement. Find the measures of both angles.

45

130

y

6y3x

35

54

x

(3x + 5) 2x

Q

x = ________

y = ________

z = ________

xz

x = ________

x = ________

y = ________

5x (3x + 4)(4x – 15) RP

x = ________

y = ________

z = ________

x = ________

x = ________

mRPQ = ________

x = ________

mBAN = ________

m RAT = ________

5x4x

x = ________

1.11 Warm Up

1. Which of the figures shown is the image of figure f when translated? Explain why or why not for each figure.

2. In the days of kings and wizards, Morganna, Merlin’s cousin, had a wondrous drawing tool as shown below. When she secured a nail in one of the instrument’s holes and placed a quill pen through the moon, she could create amazing designs. She moves the tool so that it points towards the goblet, making a mark on the surface.

a. Show the mark the pen would make.b. Draw the position of the tool when she is done drawing.

46

Foundations of Math II Due Date: ____________________

Unit 1Project

For this project, you will use the five geometric constructions that you have learned in this unit to design and draw a picture of a single object. You are expected to use at least two of each of the constructions listed below in your picture. However, don’t use too many or your picture will be cramped with construction marks.

Construction A: Congruent Segments Construction B: Congruent AnglesConstruction C: Angle Bisector Construction D: Perpendicular Line to a segment from a point on or off the segmentConstruction E: Perpendicular Bisector of a segment

You will turn in 5 items: 1) your version of the cat face that we constructed in class2) colored copy of the cat face3) the graded rough draft of your design4) final version showing construction marks5) a copy of the finished picture colored completely

Rough Draft:You should make the rough draft of your picture on a piece of unlined paper. I would recommend playing with ideas several times before submitting a rough draft to me for comments. You will need a compass, a ruler, a pencil, and colored pencil. Place the letter of the construction used next to each set of construction marks (one letter for each construction, each time it is used). Use a colored pencil in your compass for the construction marks and use a regular pencil for all marks made without the compass.

Final Construction and Picture:After I return your rough draft to you, be sure to read the comments and make corrections as you complete your final version. Next, make an identical copy of your picture by tracing it on a separate piece of paper but without any of your construction marks. Then color the picture using colored pencils.

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RUBRIC

Item Possible Points

Earned Points Comments

Cat Face – Construction drawing on unlined paper complete shows construction marks in a different

color from picture marks construction marks labeled with A-E neat

20

Cat Face – Colored Picture traced accurately colored attractively

15

Rough Draft drawing on unlined paper picture is a recognizable object shows construction marks in a different

color from picture marks construction marks labeled with A-E neat creative

25

Final Version – Construction drawing on unlined paper shows construction marks in a different

color from picture marks construction marks labeled with A-E incorporates feedback from rough draft neat creative

25

Final Version – Colored Picture traced accurately colored attractively

15

Total Points 100

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1.11 Show What You Know! Name___________________________________

Work through the following problems to review what you have learned in this unit.Part I: Points, Lines, and Planes

Part II: Distance and Midpoint

Now go back and find the midpoint for each of the odd numbered problems in Part II.49

My Score for Part I:

❑10

Mastery = 8

My Score for Part II:

❑18

Mastery = 15

Part III: Angles

Part IV: Angle Bisector

50

My Score for Part III:

❑6

Mastery = 5

My Score for Part IV:

❑6

Mastery = 5

Part V: Angle Terms

Part VI: Solving Problems involving angles.

51

My Score for Part V:

❑8

Mastery = 7

My Score for Part VI:

❑6

Mastery = 5

Part VII: Transformations

1. Find the line of reflection for the lion picture. Explain your method.

2. Find the image of triangle GHI. Be sure to label the vertices!

3. Draw the image of the teapot using the given translation arrow.

52

My Score for Part VII:

❑3

Mastery = 3

1.11 Warm Up

1. Find the pre-image of figure A’B’C’D’ that was translated using arrow f.

2. Bill rotates figure h using center V as shown by the arrow below. a. Draw and label the image of figure h. Explain your method.b. What is the image of V? Explain.


53

1.11R Show What You Know! Name_______________________________

Work through the following problems to review what you have learned in this unit. You must complete the sections that you did not show mastery of on Day 10. The other sections are optional review.

Part I: Points, Lines, and Planes

Part II: Distance and Midpoint

61

My Score for Part I:

❑9

Mastery = 7

Now find the midpoint for each segment in #1-4.

Now go back and find the midpoint for each segment in #5-8.

62

My Score for Part II:

❑16

Mastery = 13

Part III: Angles

Part IV: Angle Bisector

Part V: Angle Terms

63

My Score for Part III:

❑14

Mastery = 11

My Score for Part IV:

❑2

Mastery = 2

R

A T

Part VI: Solving Problems involving angles.

Part VII: Transformations

Draw the image of triangle RAT using the given translation arrow.

1.12 Warm Up

1. Draw figure q, the image of figure p when rotated using center V and arrow a. Explain your method.

64

My Score for Part V:

❑6

Mastery = 5

My Score for Part VI:

❑2

Mastery = 2

My Score for Part VII:

❑1

Mastery = 1

Draw figure r, the image of figure p when rotated using center V and arrow b. Explain your method.


65

Vocabulary Word

Definition CharacteristicsPicture and/or

SymbolReal Life Examples

Acute Angle

Adjacent Angles

Angle

Angle Bisector

Angle Measure

66

Vocabulary Word



Bisect

Collinear

Complementary Angles

Congruent

Congruent Angles

67

Vocabulary Word



Congruent Segments

Coplanar

Hypotenuse

Intersect

Intersecting Lines

68

Vocabulary Word



Intersecting Planes

Leg

Length of a Segment

Line

Line Segment

69

Vocabulary Word



Linear Pair

Midpoint

Noncollinear

Noncoplanar

Obtuse Angle

70

Vocabulary Word



Opposite Rays

Parallel Lines

Parallel Planes

Perpendicular Lines

Plane

71

Vocabulary Word



Point

Ray

Right Angle

Right Triangle

Segment

72

Vocabulary Word



Segment Addition Postulate

Segment Bisector

Skew Lines

Space

Straight Angle

73

Vocabulary Word



Supplementary Angles

Vertex

Vertical Angles

74

Vocabulary Word



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