chapter 1: foundations for geometry/review assignment...

Chapter 1: Foundations for Geometry/Review

Assignment Sheet

# Name Completed?

1 Video: Points, Lines Planes and Angles

2 Understanding Points, Lines and Planes

3 Measuring Segments and Angles

4 Video: Constructions 1

5 Performing Constructions 1

6 Applying Constructions 1

7 Video: Constructions 2

8 Performing Constructions 2

9 Labeling Relationships

10 Calculating Distance and Midpoint

11 The Hidden Rhombus

12 Video: Inscribed and Circumscribed Figures

13 Constructing Inscribed Figures

14 Geometry in Art (Class Activity)

15 Chapter Review

16 Chapter Test

Name ______________________________________________ Score _____/10

September/October 12

13

14

15

Pre-Requisite Test

HW: Video for

Sheet #1

16

#2 Understanding

Points, Lines and

Planes

#3 Measuring

Segments and

Angles

HW: Video for

Sheet #4

19

Skills Quiz/PSM

20

#5 Performing

Constructions 1

21

#5 Performing

Constructions 1

#6 Applying

Constructions

HW: Video for

Sheet #7

22

#6 Applying

Constructions

23

#8 Performing

Constructions 2

26

Skills Quiz/PSM

27

#9 Labeling

Relationships

28

#10 Calculating

Distance and

Midpoint

29

#11 The Hidden

Rhombus

HW: Video for

Sheet #12

30

#13 Constructing

Inscribed Figures

3

Skills Quiz/PSM

#14 Geometry in

Art (Class

Activity)

4

#15 Chapter

Review

HW: STUDY

5

CHP 1 TEST

6 7

Name __________________________________ Video: Points, Lines, and Planes

CC Geometry Chp 1 Wksht #1

Vocabulary

Point:

Line:

Line Segment:

Ray:

Angle:

Plane:

Shade XYZ :

X

Y

A

R

Z

B

C

Name __________________________________ Understanding Points, Lines and Planes


Name __________________________________ Measuring Segments and Angles


Vocabulary: Congruent: Same size and shape (equal measure)

Midpoint: Separates the segment into two congruent segments

Bisect: To divide into two equal parts

Name each of the following.

1.) The point on DA that is 2 units from D 2.) Two points that are 3 units from D

3.) The coordinate of the midpoint of AG 4.) A segment congruent to AC

5.) Use the number line below for a-d. Tell the length of each segment and whether or not the segments are

congruent.

a.) LN and MQ b.) MP and NQ c.) MN and PQ d.) LP and MQ

A B

A

C

A

D

A

E

A

F

A

G

A

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=MTACXuB1_Psu4M&tbnid=CN7gXs7E3aL65M:&ved=0CAUQjRw&url=http://www.sparknotes.com/math/algebra1/inequalities/section4.rhtml&ei=8nIXUviEKtLd4AOHzoCYBg&bvm=bv.51156542,d.dmg&psig=AFQjCNE7WJgBrvWUtBX-U0Dpjem4jS0CVA&ust=1377354863677267

Hint: When there is no picture, DRAW ONE!

For exercises 10-13, T is the Midpoint of PQ . Find the value of PT for each example.

10.) 35 xPT and 97 xTQ

11.) 64 xPT and 26 xTQ

12.) 247 xPT and 2613 xPQ

13.) oXYZ 67 and oXYR 41 find m RYZ .

14.) oXYZ 72 , 62 xXYR , 124 xRYZ , find m RYZ .

15.) YR is the angle bisector of XYZ . If 73 xXYR and 137 xRYZ , find m XYZ .

X R

Z

Y

X R

Z

Y

X R

Z

Y

Name __________________________________ Video: Constructions 1


1. Copying a segment

(a) Using your compass, place the pointer at Point A and extend it until reaches Point B. Your compass now has the measure of AB.

(b) Place your pointer at A’, and then create the arc using your compass. The intersection is the same radii, thus the same distance as AB. You have copied the length AB.

Examples

Copy the given segment.

Create the length 3AB

2. Bisect a segment

A B

A B A'

A B A'

(a) Given AB

(b) Place your pointer at A, extend your compass so that the distance exceeds half way. Create an arc.

(c) Without changing your compass measurement, place your point at B and create the same arc. The two arcs will intersect. Label those points C and D.

(d) Place your straightedge on the paper so that it forms CD . The

intersection of CD and AB is the bisector of AB .

A

B

A

B C

D

A

A'

B C

A

Examples:

1. Bisect the segment (find the midpoint). 2. Cut the segment into four equal pieces

3. Copy an angle

4. Construct a line parallel to a given line through a point not on the line.

Question: What type of angles prove these lines are parallel?

(a) Given an angle and a ray. (b) Create an arc of any size, such that it intersects both rays of the angle. Label those points B and C. (c) Create the same arc by placing your pointer at A’. The intersection with the ray is B’. (d) Place your compass at point B and measure the distance from B to C. Use that distance to make an arc from B’. The intersection of the two arcs is C’.

(e) Draw the ray ' 'A C

(a) Given a point not on the line. (b) Place your pointer at point B and measure from B to C. Now place your pointer at C and use that distance to create an arc. Label that intersection D. (c) Using that same distance, place your pointer at point A, and create an arc as shown. (d) Now place your pointer at C, and measure the distance from C to A. Using that distance, place your pointer at D and create an arc that intersects the one already created. Label that point E.

(e) Create AE .

Name __________________________________ Performing Constructions 1


COPYING A SEGMENT

1. Given , ,&AB CD EF . Use the copy segment

construction to create the new lengths listed below.

3AB

CD + EF

2CD + 1AB

EF - CD

CONSTRUCTING A MIDPOINT

2. Given AB & CD . Use the midpoint construction to find the midpoint of AB & CD

3. Use your midpoint construction to determine the exact length of 1

4EF

A BC DE F

A B

C

D

E F

4. Given ABC . Make a copy of ABC , ' ' 'A B C .

5. Given DEF . Make a copy of DEF , ' ' 'D E F .

A

B

C

E

B'

D

E

F

E'

PRACTICE - CONSTRUCTION BASICS #1

1. Given MN , construct 2.5 MN

2. Given GH , construct 1.75 GH

3. Given ABC, construct a copy of it, A’B’C’.

M N

G H

B C

A

A

B'

4. Given ABC, can you think of a way to create a line parallel to AB through point C?

(Hint: How could copying an angle help you?)

5. Create a parallel line to DE through point F.

BC

A

D F

E

D

C

A

B

Name __________________________________ Applying Constructions 1


1. Convert the mathematical symbols to words.

a) AB _______________________ b) AB _______________________

c) AB _______________________ d) AB _______________________

e) ABC _______________________ f) m ABC _______________________

2. Which geometric instrument would I use to measure the length of a segment, the compass or the straightedge?

Explain your answer.

3. What is the difference between drawing and constructing something? So for example, what is the difference

between drawing a perpendicular line and constructing a perpendicular line?

4. A student has done the following construction. What was this student attempting to construct? Is there more than

one thing that the student could be constructing? Explain.

5. After learning the midpoint construction, Sally realizes that she could determine one-fourth the length of a

segment. How could she do this? Explain & Diagram.

6. When given AB & CD , a student uses her compass to measure them and then construct a new length EF. What is

the exact length of EF ?

F

C D

A B

E

7. A teacher instructs the class to construct the midpoint of a segment. Jeff pulls out his ruler and measures the

segment to the nearest millimeter and then divides the length by two to find the exact middle of the segment. Has he

done this correctly?

8. What is the difference between CD and CD ?

9. Given three possible correct names for the given angle.

10. When do we use = and when do we use ?

11. What does it mean to bisect something?

12. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and

then explain why it MUST be a rhombus.

1C

A

B

D

C

A

B

A

Name __________________________________ Video: Constructions 2


1. Construct the perpendicular bisector of a line segment

2. Construct a line perpendicular to a given segment through a point on the line.

Example: Construct the perpendicular line through a point on the line.

(a) Given AB

(b) Place your pointer at A, extend your compass so that the distance exceeds half way. Create an arc.

(c) Without changing your compass measurement, place your point at B and create the same arc. The two arcs will intersect. Label those points C and D.

(d) Place your straightedge on the paper and create CD .

(a) Given a point on a line.

(b) Place your pointer a point A. Create arcs equal distant from A on both sides using any distance. Label the intersection points B and C.

(c) Place your pointer on point B and extend it past A. Create an arc above and below point A.

(d) Place your pointer on point C and using the same distance, create an arc above and below A. Label the intersections as points D and E.

(e) Create DE .

A

B

A

3. Construct a line perpendicular to a given line through a point not on the line.

Example: Construct the perpendicular line through a point not on the line.

4. Bisect an angle

Example: Bisect the angle

(a) Given a point A not on a line.

(b) Place your pointer a point A. Create arcs equal distant from A on both sides using any distance. Label the intersection points B and C.

(c) Place your pointer on point B and extend it past A. Create an arc below point A.

(d) Place your pointer on point C and using the same distance, create an arc below A. Label the intersections as points D.

(e) Create DE .

(a) Given an angle.

(b) Create an arc of any size, such that it intersects both rays of the angle. Label those points B and C.

(c) Leaving the compass the same measurement, place your pointer on point B and create an arc in the interior of the angle.

(d) Do the same as step (c) but placing your pointer at point C. Label the intersection D.

(e) Create AD . AD is the angle bisector.

A

Name __________________________________ Performing Constructions 2


Constructing the Perpendicular Bisector (a line through the midpoint of a segment).

1. Given AB . Use the midpoint construction to construct the perpendicular bisector.

Construct the perpendicular line THROUGH A POINT ON THE LINE. 2. Work backwards from the midpoint construction.

Construct the perpendicular line THROUGH A POINT not on THE LINE. 3. Work backwards through the midpoint construction.

A B

A

B

AC

B A

Construct the angle bisector.

4. Given A, construct the angle bisector, ray AD .

5. Given sides of a rectangle. Construct the rectangle. Hint - We need perpendicular lines through A and through M.

6. Given the side of a square. Construct the square.

A

A

A B

C

D

Name __________________________________ Labeling Relationships


1. Use the diagram to complete the relationship.

a) ________= ________

b) ________ ________

c) ________= ________

d) ________ ________

e) ________= ________

f) ________ ________

g) ________= ________

h) ________ ________

2. Choose which construction matches the diagram.

a) The Midpoint of BC

b) line through A

c) bisector d) Copy a segment

a) Copy

b) bisector


a) Copy

b) bisector


a) The Midpoint of BC

b) line through A


3. A rhombus is a quadrilateral with 4 congruent sides. Hidden in this construction is a rhombus, can you find it and

then explain why it MUST be a rhombus.

4. If you are told that MN is the perpendicular bisector of BC where point M is on BC . Draw the diagram and

completely label it with all known relationships.

5. If you are constructing the perpendicular line through point A (A is on the line), determine the next step.

Step #1 – Place compass at point A, and create two intersections B & C on either side of point A.

Step #2 – Place compass pointer at point B and extend its measure beyond A and make an arc above

and below point A.

Step #3 -- __________________________________________________________________________

M

A

Bo

oD

C

A

B

CB

A

D

CB

A

D

C

A

B C'

B'

C

A

A'

B

C A B

D

C

A

B

Construct a perpendicular line through point B. On the right side, list all the steps of the construction

and label any relationships on the diagram.

Construct the midpoint of segment AB. On the right side, list all the steps of the construction and label

any relationships on the diagram.

Construct the angle bisector of angle A. On the right side, list all the steps of the construction and label

any relationships on the diagram.

B

A

B

A

Name __________________________________ Calculating Distance and Midpoint

NR Geometry Chp 1 Wksht #10

Formulas:

Midpoint Distance

2,

2

1212 yyxx 2

12

2

12 )()( xxyyd

Use the distance formula to determine the distance between each pair of points. Answer in simplified radical form

when appropriate.

1) (3, 10) and (–4, 31)

2) (–2, –12) and (4, –9) 3) A(3, 4) and B(-2, 4)

4.) 5.)

6.) 7.)

8.) 9.)

Name __________________________________ The Hidden Rhombus


Why do these constructions work?

Construct a Midpoint Construct a Perpendicular Line

Through a Point Not on the Line Construct a Perpendicular Line Through a

Point On the Line

Construct an Angle Bisector

1. When performing these constructions you kept your compass at a fixed length. This creates distances that are equal - thus the 4 equal sides of a rhombus. a) Use a ruler and draw in the missing sides of the rhombus in each of these constructions. b) Shade the rhombus with your pencil.

2. In the first three constructions we needed to create lines that were perpendicular.

Which property of a rhombus do you think we are using to accomplish this goal?

Opposite sides are || Diagonals bisect each other

Opposite sides are Diagonals are

Opposite angles are Diagonals are Bisectors

Consecutive ’s = 180

3. In the angle bisector construct we need to divide an angle into two equal parts.

Which property of a rhombus do you think we are using to accomplish this goal?

Opposite sides are || Diagonals bisect each other

Opposite sides are Diagonals are

Opposite angles are Diagonals are Bisectors

Consecutive ’s = 180

4. Given VB -- perform the midpoint construction. This time labeling the two intersection found to be H and K. Draw in

, , ,& .VH VK BH BK Also draw HK .

Why is VH = VK? _______________________________________________________________________

Why is BH = BK? _______________________________________________________________________

Why is VH = VK = BH = BK? _______________________________________________________________

What is the most specific name for the quadrilateral VHBK? _____________________________________

Will this specific quadrilateral be formed every time using this construction? Yes or No

Why or why not…

Label the intersection of HK and VB is point M.

What is true about VM and BM? _______________________________________________________

What is true about HM and KM? _______________________________________________________

What is the measure of the angle formed at the intersection of HK and VB ? _________________

V B

Name __________________________________ Video: Inscribed and Circumscribed Figures


1. The construction of an inscribed equilateral.

Example:

Construct an inscribed Equilateral, label all congruent parts.

2. The construction of an inscribed square.

Example: Construct an inscribed Square.

(A) Given Circle A

(B) Create a diameter BC

(C) Create a circle at C with radius AC . Label the two intersections D and E.

(D) Create BD , BE & ED

(A) Given Circle A


(C) Construct a perpendicular line to BC through A.

(D) Create BD , DC , CE & EB

A

A

3. The construction of an inscribed hexagon.

Example:

Construct an inscribed Hexagon.

Inscribed vs. Circumscribed

Inscribed: Circumscribed

(A) Given Circle A


(C) Create a circle at C with radius AC . Label the two intersections D and E.

(D) Create a circle at B with radius BA . Label the two intersections F and G.

(E) Create CD , DF , FB , BG , GE & EC

A

Name __________________________________ Constructing Inscribed Figures


1. Determine whether the relationship is INSCRIBED or CIRCUMSCRIBED.

a) The triangle is _____________. b) The hexagon is _____________ c) The circle is _______________

d) The hexagon is _____________ e) The circle is _______________ f) The triangle is ______________

2. Jeff uses his compass to make a cool design. He just keeps creating congruent circles… over and over…

a) Find a regular hexagon (shade it in) b) Find a different regular hexagon (shade it in)

c) Find an equilateral triangle (shade it in) d) Find a different equilateral triangle (shade it in)

3. The inscribed equilateral triangle has a central angle of 120 because 360 / 3 = 120, an inscribed square has a

central angle of 90 because 360 / 4 = 90. The central angle of a decagon is 36 because 360 / 10 = 36. Use this

information and a compass to create an inscribed decagon.

4. Construct the requested inscribed polygons.

a) Construct an equilateral triangle inscribed in the provided circle using your compass and straightedge.

b) Construct a square inscribed in the provided circle using your compass and straightedge.

36°

5. Construct the requested inscribed polygons.

a) Construct a regular hexagon inscribed in the provided circle using your compass and straightedge.

b) Construct a regular octagon inscribed in the provided circle using your compass and straightedge.

Hint: The central angle is 45, half of

the square’s central angle of 90.

Name __________________________________ Geometry in Art


Now that you know the basics of constructions you have the ability to create some very cool Geometric Art. Using just the construction of the inscribed hexagon (and the addition of a few segments and arcs) you can create a number of very interesting designs. Here are a few examples of Geometric Art. (The QR code links to an introduction video.)

Practice Area (Shading & coloring change the look of the shape…. Also symmetry is very appealing to the eye.)

Final Copy (Full Color & Clarity) Name: ____________________ Period: ____

Name __________________________________ Chapter 1 Review


Know the following definitions and any appropriate symbols (ie segment AB = AB )

Point

Plane Line Line Segment Ray Angle

Congruent

Perpendicular Supplementary Complementary Bisect Bisect

Collinear

Noncollinear Coplanar Noncoplanar Adjacent Vertical Angles

Midpoint

Distance Acute Obtuse Straight Angle Right Angle

Area Supplement Complement

Review the basic constructions.

1. Copying Segments a) 4AB – CD

b) 2.75CD + AB

2. Midpoint Construction

a) Given diameter AB, find the center

b) Which is larger AB or ½ CD? ____________

A B C D

B

A

Copy them here to compare them

A

B

C

D

c) Construct the midpoint of AB

d) 1

3CD GH . Construct GH .

3. Copying an Angle & Parallel Line Construction

a) Create A’B’C’ by copying ABC.

b) Create D’E’F’ by copying DEF.

c) Create A’ by copying Copy A d) Create E’ such that mE’ = 2(mE)

A

B

C

D

G

B

A

C

B'

E

D

F

E'

A

A'

E

E'

4. Angle Bisector Construction

a) Construct the angle bisector of A. b) Create DBC such that mDBC = ½ (mABC)

5. Perpendicular Constructions Construct the following.

a) the perpendicular bisector of AD b) a perpendicular line to AE through E

c) the perpendicular line to DA through C

6. Inscribed Polygons

a) Inscribed Square b) Inscribed Equilateral Triangle c) Inscribed Hexagon

A

B

A

C

A

C

E

D

7. Labeling and Communicating Relationships.

a.) Perform the perpendicular bisector construction. Label all relationships on the diagram and list the steps on

the right.

b.) Julia is constructing a perpendicular through a point on the line. What is the next step? Be very specific as to

what he should do next.

1. Given a point on a line.

2. Place your pointer a point A. Create arcs equal

distant from A on both sides using any distance.

Label the intersection points B and C.

3. Place your pointer on point B and extend it past

A. Create an arc above and below point A.

4.

A B

C A B

chapter 1: foundations for geometry/review assignment...

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