Unit 1 : Intro to GeometryNotes & Homework

Name:

Ms. Ferreras Fo Math II -Fall 2016

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Foundations Math II Name_____________________ Pd ____Stamp Sheet

Unit 1: Introduction to Geometry

Day Lesson Homework Stamp1 Basic Geometric Vocabulary & Segment

Addition Postulate

2 3-D Drawing & Algebra Review

3 Pythagorean Theorem & 3-D Shapes

4 Mira Labs and Segment Addition

5 Quiz

6 Labor Day - No School

7 Distance, Midpoint, & Tangrams

8 Angle Addition Postulate

9 Quiz

10 Construction Video Practice Day

11 Study Island & Foldable Const. Day

12 Project Plan Day

13 Project Presentation Day

14 Unit Test 1

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Unit 1: Intro to

GeometryMain ConceptsOnline PracticeMath Warehouse – Geometry

1. Definitions Point, line, plane 2. Segment Addition Postulate 3. Pythagorean Theorem4. Drawing 3-D Shapes5. Distance and Midpoint Formula6. Geometric Constructions: Construction A: Congruent Segments Construction B: Congruent Angles Construction C: Angle Bisector Construction D: Perpendicular Line to a segment from a point on or off the segment Construction E: Perpendicular Bisector of a segment

Vocabulary List

Angle Bisector Congruent Segments Midpoint Segment Bisector

Angle Measure Hypotenuse Obtuse Angle Straight Angle

Bisect Intersecting Planes Opposite Rays Supplementary Angles

Complementary Angles Leg Parallel Planes Vertex

Congruent Length of a Segment Right Angle Vertical Angles

Congruent Angles Linear Pair Right Triangle

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Know a little Need Practice

I Got it!

Day 1 - LessonState whether each is best modeled by a point, line, or plane.

1. a knot in a piece of thread 2. a piece of cloth3. the corner of a room4. the telecommunications beam to a satellite in space5. the crease in a folded sheet of wrapping paper6. an ice skating rink

Create a definition of what makes a point, line, and a plane below:Point

Line

Plane

List two to three real-world objects that could be modeled using a point, a line, and a plane. Point

Line

Plane

Basic Vocabulary Definition Picture Symbol/Notation

PointUndefined A point has no size, no

dimension.It is a location in space. Points are named using

A

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capital letters.

LineUndefined A line is made up of an

infinite number of points that go on without end in both directions.

A line has one dimension, length.A line can be named with a single lowercase script letter or by two points on the line.

AB

l

PlaneUndefined A flat surface that extends

without end in all directions.

It has two dimensions, length and width.

A plane can be named by three non-collinear points or by a single uppercase script letter.

A parallelogram is used to model a plane.

RayPart of a line

A ray has a definite starting point and extends without end in one direction.

A ray is named using the endpoint first, then another point on the ray.

AB

Line SegmentA line segment is part of a line containing two endpoints and all points between them.

A line segment is named

AB

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using the endpoints.

CollinearPoints that lie on the same line.

Non-collinearPoints that do not lie on the same line.

CoplanarPoints that lie in the same plane.

.

Non-coplanarPoints that do not lie in the same plane.

Space Boundless, three-dimensional set of all points

Parallel linesLines in the same plane that are always the same distance apart.They do not intersect.

Perpendicular LinesLines that intersect to form right angles

Skew LinesLines that are not in the sameplane and do not intersect.

IntersectionThe set of points that are in both figures.

Concurrent linesThree or more lines that

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pass through the same point.

Postulates: (Statements that are accepted as true without proof).

Two points determine a unique line.

If two distinct lines intersect, then their intersection is a point.

Three non-collinear points determine a unique plane.

If two distinct planes intersect, then their intersection is a line

Dihedral Angle: The angle formed when two planes meet

Segment Addition Postulate

Segment Addition Example

Example 1:

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Name a point that is collinear with the given points_ B and E:_ C and H:_ D and G:

Example 2:

Determine if the pairs of lines appear to be parallel, perpendicular, or skew. It’s OK to write symbols for parallel and perpendicular.

a) Line BF and line AE are ________________ b) Line CG and line HG are ______________c) Line AD and line AE are _______________ d) Line FG and line EH are ____________e) Line AD and line HG are _______________ f) Line HG and line BF are _______________

Day 1 - HW8

1.

Name a point that is coplanar with the given points_ M, N, R:_ M, N, O:_ M, T, Q:

2. Use the diagram below.

a. Name the intersection of line BE and line CD .

b. Name the intersection of line BE and plane A.

c. Name the intersection of line DB and plane F.

d. Name the intersection of plane A and F.

3. In the figure below, B is the midpoint of AC. Find x and AC.A 2x + 12 B 5x +10 C

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

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5. In the figure below, C is the midpoint of AB. Find x and AC. 5x – 6 2x

A C B

6. If RT = 60, find x, RS, and ST. R 3x – 12 S 2x – 8 T

Multiple Choice

7. The intersection of 2 line segments can be:

I. a pointII. a line segmentIII. a rayIV. a line

A. I only B. I and IIC. I and IV D. all of the above

9. The intersection of 2 distinct lines can be:

I. a pointII. a line segmentIII. a rayIV. a line

A. I only B. IV onlyC. I, III and IV D. all of the above

8. The intersection of 2 distinct rays can be:

I. a pointII. a line segmentIII. a rayIV a line

A. I only B. IV onlyC. I, II and III D. all of the above

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Day 2 - Lesson

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Solve for x and y. −20=−4 x−6 x 2 (4 x−3 )−8=4+2x

−4 (−6 x−3 )=12 4 y−2=3 y+1 y

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

1. point: 2. line:3. plane:4. space:

I. 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

II. 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?

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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.

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?

III. Thinking back. Use your answers to Part II 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!)

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Day 2 - HWDraw 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.

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 .

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E

H

TM

V

O

L

A

Day 3 - Class work: Pythagorean Theorem

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

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

2. A baseball diamond is a square with a side length of 90 feet. To the nearest foot, what is the distance between first base and third base?

2nd base

3rd base 1st

base

Home plate

A 810 ft

B 90 ft

C 180 ft

D 127 ft

3. A flower garden is in the shape of a right triangle. One leg of the triangle is 6 ft long and the hypotenuse is 12 ft long. What is the length of the other leg?

A 14.6 ft

B 10.4 ft

C 13 ft

D 12.8 ft

4. Which of the following sets of numbers represents a Pythagorean Triple?

A { 10, 5, 12}

B { 2, 4, 5}

C {1, 2, 3 }

D {14, 48, 50 }

5. find the missing length

a = 7, c = 25, b = ?

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3 – D Shapes in Power Point ActivityCreate two different 3 – D shapes using Power Point.State the location of 4 different vocabulary words on each shape. See the list below:

PointLinePlaneLine segmentParallel LinesPerpendicular linesSkew LinesIntersection

Follow the directions from the two Youtube video clips below (See Webpage Class Activities Links 1 & 2):

https://www.youtube.com/watch?v=Wm8sxxU8jFk

https://www.youtube.com/watch?v=BO981ngojEo

Assignment due at the end of the period.

3 – SlidesTitle – shape 1 & vocab – shape 2 & vocab

Save results as a ppt & pdf. Put the pdf in the Google Doc on the website.

Day 3 - HW

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1. A 12-foot ladder is leaning against the side of a building. The top of the ladder reaches 10 feet up the side of the building. Approximately how far is the bottom of the ladder from the base of the building?

A 2 ft

B 15.6 ft

C 6.6 ft

D 1.4 ft

2. Which of the following sets of numbers does NOT represent a Pythagorean Triple?

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A {34, 16, 30}B {10, 24, 26}C {19, 21, 23}D {5, 12, 13}

3. A right triangle has legs of lengths 9 cm and 12 cm. What is the length of the hypotenuse?

A 15 cmB 22 cmC 25 cmD 14 cm

4. A triangle has legs which measure 15 inches and 20 inches. Find the length of the hypotenuse.

A 28 inchesB 32 inchesC 13 inchesD 25 inches

5. A 20-ft-long wire is used to support a television antenna. The wire is connected to the antenna 15 ft above the ground. How far away from the base of the tower will the other

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end of the wire be located?

A 13.2 ftB 25 ftC 18 ftD 16.5 ft

6. Gina is making a quilt in the shape of a right triangle. The measurement of the diagonal of the quilt is 36 in. long. The width of the quilt is 15 in. long. What is the approximate length?

A 27.3 in.B 32.7 in.C 19.6 in.D 29.9 in.

7. Which set of squares could form a right triangle?

Day 4: Lesson – 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 “math tool” 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!)

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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.

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

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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!

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!)

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Day 4 - HW (# 1 - 24)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 = ?

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T S V

B

Q W R

P

A

BC

X

DE

F

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

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

17. What does a Mira or a mirror have to do with Geometry?

Challenge:

18. 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.

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

19. a star in the sky

20. the floor of the gym

21. an overhead power line

22. a pixel on a computer screen

23. the top of your school desk

24. the spine of a book

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T B W

D

O

TC

G

Day 5 (QUIZ DAY)

HWUse symbols to name each of the following objects in picture to the right.1. a line containing point W2. two points collinear with point X3. a line passing through point Y4. two points coplanar with point Z5. a point not on plane R.6. Challenge: two points coplanar with point U

Using colored pencils, label and shade the figure as described.7. With a yellow pencil, shade a plane. Then label three

noncollinear points on the plane as R, S, and T.8. With an orange pencil, shade a plane that intersects

the yellow plane.9. What is the intersection of the orange and yellow

planes?10.With a red pencil, label four coplanar points as E, F, G,

and H.11.With a blue pencil, label three collinear points as W, X,

and Y.

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

Use the number line to find each measure.13. HK 14. LN 15. LJ 16.

KM

Day 7 - LessonUse the Distance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from D(3, 4) to E(–2, –5).

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NMLKJIK

H

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

Determine whether FG JK.

Find the coordinates of the midpoint of PQ with endpoints P(–8, 3) and Q(–2, 7).

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S is the midpoint of RT. R has coordinates (–6, –1), and S has coordinates (–1, 1). Find the coordinates of T.

M is the midpoint of XY. X has coordinates (2, 7) and M has coordinates (6, 1). Find the coordinates of Y.

Day 7 – HW (#1 – 10 & Tangram Activity)

Use the Pythagorean Theorem to find the distance between each pair of points.1. 2.

3. K (-2, -6), L(3, 6) 4. S(-3, -1), T(1, 5)

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Use the number line to find the coordinate of the midpoint of each segment.

5. KM 6. KN 7. HJ 8. JL

Find the coordinates of the midpoint of the segment with the given en

9. B(3, 1), C(5, 5) 10. W(-4, 3), V(5, -3)

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NMLKJIH

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

HW Continued: Constructing a TangramA 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.

Graph Paper for Tangrams

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Day 8 – Lesson

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Day 8: HW (# 1 – 6) 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

Find the distance and midpoint between each pair of points. 5 (4, 6), (-2, -1) 6. (-3, -4), (2, 0)

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R

QS

P

X

T

Day 9: Lesson (Quiz Day)

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Day 9 - HW (# 1 – 3)Group Activity # 1

1. 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 = ________

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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 + 11

m13 = 3x + 1

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17m16 = 5x + 3

m17 = 9x – 5

16 and 17 are supplementary

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

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

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

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L

O

NM

PQ

QM bisects LQN

QO bisects PQN

Day 10 (Construction Video) – HW

Day 11 & 12 (Foldable): HW Practice Test Use the figure at the right for #1-7. G is the midpoint of RS and <RGK ¿<KGB.

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1. Name a ray opposite to DS ________2. Name two congruent segments. __________3. Name an angle bisector. ______________4. Because of this postulate ________________,

AG + GB = _______________.5. m<RGK _______ m<KGB.6. m<RDC + m< ________ = 180.7. If RG = 4, than RS = ___________.

Use the figure at the right for #8-11.

8. Name the intersection of plane M and N. __________9. Are points A, B, and D collinear? ____________10.Are points A, B, and D coplanar? ____________11.Name two points that determine line l. _________

Use the figure at the right for #12-16. 12.Name a plane that contains EF. ________13.Name two planes that do not intersect. _________14.Name three lines shown that intersect at point H.

_________________15.Name a fourth point that is in the same plane as the

points D, E, F. _________16.Name the intersection of planes AEHD and EFGH.

_________________

Sketch and label the figures described.

17.Vertical line k intersects horizontal plane R at C.

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18. Q is the midpoint of JK.

19.Plane N contains intersecting lines r and s.

20.J, K, and L are coplanar points but not collinear points.

Classify each statement as true or false and give the definition, postulate, and theorem that supports your conclusion.

21.Two intersecting lines meet in exactly one point. ___________________________22.A given triangle can lie in more than one plane. _____________________________23.Two planes can intersect in only one point. _________________________________24.Three points determine a plane. _________________________________________25.Through a line and a point not on the line there is one and only one plane.

__________26.Congruent segments are segments that sometimes have equal measures.

___________27.If S is between R and T, than R, S, and T are collinear points.

___________________28.Use the figure below to find the value of x.

29.DB bisects <ADC, m<1 = 5x – 3, and m< 2 = x + 25. Find the value of x.

30.E is the midpoint of DF. DE = 45 and EF = 5y – 10. Find the value of y.

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Day 13 – LessonFoundations of Math 2 – Geometric Construction “Cat” Project Rubric

The goal of this project is to creatively demonstrate your ability to follow directions and create a geometric picture using a compass and ruler.

1. Create a “Clean and Legal” Cat face using the following five geometric constructions:o Construction A: Congruent Segments o Construction B: Congruent Angleso Construction C: Angle Bisector o Construction D: Perpendicular Line to a segment from a point on or off the

segmento Construction E: Perpendicular Bisector of a segment

at least two of each of the constructions listed above must be in your picture.2. Each project must contain the following 5 items:

o your version of the cat face that we constructed in classo colored copy of the cat faceo the graded rough draft of your designo final version showing construction markso a copy of the finished picture colored completely

3. Grading Scale: Good Luck & Have Fun!

Rough Draft:o Make the rough draft of your picture on a piece of unlined

paper.o You will need a compass, a ruler, a pencil, and colored pencil.

o Place the letter of the construction used next to each set of construction marks (one letter for each construction, each time it is used).

o 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:o The assignment may be turned in early, for teacher comments and for the

student to make corrections before the completed final version is turned in for grading purposes.

o Next, make an identical copy of your picture by tracing it on a separate piece of paper but without any of your construction marks.

o Then color the picture using colored pencils.

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RUBRIC Name __________________________________________

ItemPossible

PointsEarned 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

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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

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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

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Final Version – Colored Picture traced accurately colored attractively

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Total Points 100

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