unit 1b modeling with geometry:...

Common Core Math 2 Unit 1A Modeling with Geometry

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Name:____________________________ Period: _____ Estimated Test Date: ___________

Unit 1B

Modeling with Geometry:

Congruency


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3

Main Concepts Page #

Study Guide 4

Vocabulary 5-7

Similarity 8-14

Congruency 15=19

Triangle Theorems 20-23

CPCTC 24-26

Cross-Sections 27-29

Partitioning a Line Segment 30-33

Unit Review 34-36

HW Answers 37-28


4

Common Core Standards

G-CO.7 Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent.

G-CO.8 Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

G-CO.10 Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

G-CO.13 Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

G-STR.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and

leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Unit Description In this unit, students study the concept of congruency and explore sufficient conditions for congruent triangles. Students are also introduced to proof, informally proving the Triangle Angle Sum Theorem and the Midsegment Theorem. Students apply geometry concepts to modeling situations to solve problems.

Essential Questions By the end of this unit, I will be able to answer the following questions:

Why are only four types of transformations needed to describe the motion of a figure?

How do the SSS, SAS, and ASA congruence criteria follow from the definition of congruence in terms of rigid motions?

Unit Skills I can Congruent Figures: Use the definition of congruence in terms of rigid motions to decide if two figures are congruent. Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and

only if corresponding pairs of sides and corresponding pairs of angles are congruent. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence

in terms of rigid motions.* Show that the AAA and SSA criteria are not sufficient to prove that two triangles are congruent.* Triangles: Prove the Triangle Angle Sum Theorem. Draw the midsegment of a triangle. Prove the Midsegment Theorem. Applications: Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. Find the point on a directed line segment between two given points that partitions the segment in a given

ratio. Modeling: Identify the shape of a cross-section of a three dimensional object. Identify the three dimensional figure generated by the rotation of a two-dimensional shape. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk

or a human torso as a cylinder). Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile,

BTUs per cubic foot). Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios).


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Vocabulary: Define each word and give examples and notes that will help you remember the word/phrase.

AA Similarity

Example and Notes to help YOU remember:

AAS Theorem


ASA Postulate


Congruent


CPCTC


Cross-Section



6

HL Theorem


Isosceles Base Angle Theorem


Midsegment Theorem


PAP Similarity


PPP Similarity


SAS Postulate



7

Similarity


SSS Postulate


Triangle Sum Theorem



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Properties of Dilations/Similarity

Dilate ∆𝐴𝐵𝐶 about the origin with a scale factor of 2. Graph the new triangle; label the vertices 𝐴′, 𝐵′, & 𝐶′. Complete the following using your dilation.

1. Using a protractor, measure the angles of ∆𝐴𝐵𝐶 and ∆𝐴′𝐵′𝐶′. What do you notice?

2. Using the distance formula, calculate the lengths of 𝐴𝐵, 𝐴′𝐵′, 𝐴𝐶, 𝐴′𝐶′, 𝐵𝐶, 𝑎𝑛𝑑 𝐵′𝐶′. What do you

notice?

3. Dilations create similar figures. Based on your observations from 1 and 2, what can we say about similar

figures?

4. What do you notice about 𝐴𝐵̅̅ ̅̅ 𝑎𝑛𝑑 𝐴′𝐵′̅̅ ̅̅ ̅̅ ? 𝐴𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝐴′𝐶′̅̅ ̅̅ ̅̅ ? Note that A and A’ lie on the origin. What

conclusion can you make about the segments of an image when the corresponding segments of the

preimage pass through the center of dilation?

5. Using the slope formula, calculate the slopes of 𝐵𝐶̅̅ ̅̅ 𝑎𝑛𝑑 𝐵′𝐶′̅̅ ̅̅ ̅̅ . What do you notice? What conclusion

can you make about the segments of an image when the corresponding segments of the preimage do

not pass through the center of dilation?

A

B C


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Similarity

Similar Triangles that have the same _____________ but not necessarily the same _____________.

If

____ ____ , ____ ____ , ____ ____

and

= =

Prove Similarity by

________ Similarity Theorem: 3 pairs of ______________ ________

________ Similarity Theorem: _____ pairs of proportional sides and __________________ angle between

them

ABC DEF

A

B C

E

F D


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________ Similarity Theorem: 2 pairs of ____________ __________

Definition: Triangles are _________________ () if the corresponding ______________ are ____________ and

the __________________ of the lengths of the corresponding sides are __________________.

Practice

Given that the triangles are similar, determine the scale factor and solve for the missing lengths.

a. ∆𝑉𝑈𝑊~∆𝐾𝑈𝐿 b. ∆𝐿𝐵𝑀~∆𝐶𝐵𝐷 c. ∆𝑅𝑄𝑃~∆𝐷𝐶𝐵

G

H I

L

J K

M

N O

Q

P R


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Similarity HW (©Kuta Software LLC. All rights reserved.)


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CW/HW Directions: Answer the following problems deal with congruency and rigid motion.

1. In the diagram at the right, a transformation has occurred on ABC.

a) Describe a transformation that created image ABC from ABC.

b) Is ABC congruent to ABC? Explain.

2. The vertices of MAP are M(-8, 4), A(-6, 8) and P(-2, 7).

The vertices of MAP are M(8, -4), A(6, -8) and P(2, -7).

a) Plot MAP.

b) Verify that the sides of the triangles are congruent by using the distance formula.

c) Describe a rigid motion that can be used to MAP

3. Given PQR with P(-4, 2), Q(2, 6) and R(0, 0) is congruent

to STR with S(2, -4), T(6, 2) and R(0, 0).

a) Plot STR.

b) Describe a rigid motion which can be used to verify the triangles are congruent.


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4. Given RST with R(1, 1), S(4, 5) and T(7, 5).

a) Plot the reflection of RST in the y-axis and

label it RST.

b) Is RST congruent to RST? Explain.

c) Plot the image of RSTunder the

translation (x, y) (x + 4, y – 8). Label the image of RST.

d) Is RST congruent to RST? Explain.

e) Is RST congruent to RST? Explain.

5. Given DFE with D(1, -1), F(9, 6) and E(5,7) and BAT with B(1, 1), A(-6, 9) and T(-7, 5).

a) Describe a transformation that will yield BAT as

the image of DFE.

b) Is BAT congruent to DFE? Explain.

6. Given CAP with C(-4, -2), A(2, 4) and P(4, 0) and SUN with S(-8, -4), U(4, 8) and N(8, 0).

a) Describe a transformation that will

yield SUN as the image of CAP.

c) Is CAP congruent to SUN? Explain.


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Congruent Triangles Investigation What does it mean to say two triangles are congruent?

Part 1: List the ways to justify that triangles are similar.

Examine the triangles with all side lengths labeled.

a. Are they similar? Why?

b. What is the scale factor? _______ : _______

c. What do we know about the corresponding angles of similar triangles?

d. What does this tell us about the pair of triangles?

Part 2: Examine the triangles with two sides lengths and an included angle labeled.



c. Since the triangles are similar, what do we know about P and H? What do we know about I and

O?

d. Use the scale factor you gave in part b to determine the length of OH.

e. What does this tell us about the pair of triangles?

3 cm2 cm

4 cm4 cm

3 cm2 cm

T

R

SB

C

A

1.5 cm

5 cm

4 cm

4 cm

5 cm H

O

G

I

P


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Part 3: Examine the triangles with two angle pairs marked congruent.



c. Since the triangles are similar, what do we know about K and Y?

d. Use the scale factor you gave in part c to determine the lengths of KL and YZ.

e. What does this tell us about the pair of triangles?

CONGRUENCY POSTULATES AND THEOREMS

Think back to the three situations we examined.

In Part 1, we were given 3 pairs of sides of one triangle are congruent to 3 pairs of sides of another triangle.

The triangles are congruent by the ___________, __________, __________ (SSS) Postulate.

In Part 2, we were given 2 pairs of sides and an included angle of one triangle are congruent to 2 pairs of

sides and an included angle of another triangle. The triangles are congruent by the

___________, __________, __________ (SAS) Postulate.

4 cm

2.15 cm

4 cm

3.2 cm

32523252

K

L M ZX

Y

3 cm2 cm

4 cm4 cm

3 cm2 cm

T

R

SB

C

A

1.5 cm

5 cm

4 cm

4 cm

5 cm H

O

G

I

P


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In Part 3, we were given 2 pairs of angles and an included side of one triangle are congruent to 2 pairs of

angles and an included side of another triangle. The triangles are congruent by the

___________, __________, __________ (ASA) Postulate.

Two other Postulates

Angle, Angle, Side Theorem

Which Postulate is used to create this theorem?

Hypotenuse, Leg Theorem

By Pythagorean Theorem, if you know the hypotenuse and 1 leg, you can calculate the 2nd leg by

_________________ Theorem and prove congruence by ____________ postulate.

Note:

Postulate – Statement which is taken to be true without proof.

Theorem – Statement that can be demonstrated to be true by accepted mathematical operations and arguments.

4 cm

2.15 cm

4 cm

3.2 cm

32523252

K

L M ZX

Y

5 cm 5 cm 10 cm 10 cm


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HW SSS, SAS, ASA, AAS, and HL Congruence (©2011 Kuta Software LLC. All rights reserved.)


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Isosceles Triangle Investigation 1. In the box, draw an angle and label the

vertex C. This will be your vertex angle.

Measure C.

2. Using point C as center, swing an arc that

intersects both sides of C

3. Label the points of intersection A and B. Construct side AB. You have constructed isosceles ΔABC with base AB.

4. Measure sides AC and BC. What is the relationship between AC and BC?

5. Use your protractor to measure the base

angles (A and B) of isosceles ΔABC.

6. Compare your results with the rest of the class. What relationship do you notice about the base angles of each isosceles triangle?

Midsegment investigation

1. Draw triangle JKL with points J (-2, 3), K (4, 5) and L (6, -1)

2. Find the midpoint M on JL and N on KL using the midpoint

formula.

3. Find the slope of lines JK and MN. What is the relationship

of the slopes? What does this prove about JK and MN?

4. Find the length of JK and MN using the distance formula.

What is the relationship between JK and MN?

5. What do you think you can generalize about the midsegment of a triangle?

Triangle Sum Theorem investigation


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1. Using a protractor, measure each of the interior angles in both triangles. Record your findings below:

mMBP = _______ mAGO = _______

mBPM = _______ mGOA = _______

mPMB = _______ mOAG = _______

2. What is the sum of the three interior angles of each triangle?

3. What can you generalize about all triangles?

Isoceles Base Angle Theorem

Isosceles Base Angle Theorem Converse

Triangle Midsegment Theorem

Triangle Angle Sum Theorem


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HW Triangle Sum Theorem, Isosceles Triangle Theorem & Converse, Midsegments Find the values of the variables. You must show all work to receive full credit. Figures are not drawn to scale. 1. x = _____ y = _____ z = _____ 2. x = _____ 3. x = _____ 4. x = _____ 5. x = _____ 6. x = _____ 7. x = _____ y = _____ 8. x = _____ y = _____

9. Find the perimeter of ABC.

x y

z 10

20

8

x 6

6

4

4

13

10

x

5

x + 1

2x

31 5

60°

x

6x + 11

x + 23 3y - 9

2y + 6

3x - 6

x 2x + 1

y

26

12

23 A

B

C


23

50

x

x 54

63

11

10x

2x + 75x - 8

4040

F G

4x - 6

18

16

F

G H

x

98

x

74

12

10

10

A D

E

B C

(10x+6)

(5x -10)

(4x + 20)

10. One side of the Rock and Roll Hall of Fame is an isosceles triangle made up of smaller triangles based on

mid-segments. The length of the base of the building is 229.5 feet. What would the base of the bold

triangle be?

Find the value of x. 11. 12. 13. 14. x = ________ x = ________ x = ________ x = ________ 15. 16. 17. 18. x = ________ x = ________ x = ________ x = ________ 19. 20. 21. 22. x = ________ x = ________ x = ________ x = ________ Complete the following using the diagram to the right.

23. a. If 𝐸𝐴̅̅ ̅̅ ≅ 𝐸𝐷̅̅ ̅̅ , then _______ _______.

b. If 𝐸𝐵̅̅ ̅̅ ≅ 𝐸𝐶̅̅̅̅ , then _______ _______.

c. If ΔEAD is an isosceles right triangle with right angle AED, then the measure of A is ________.

229.5 ft

10x+5 7x+20

x


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Corresponding Parts of Congruent Triangles are Congruent

_________________ is an abbreviation for the phrase “Corresponding Parts of Congruent Triangles are

Congruent.”

SSS, SAS, ASA, AAS, and HL use corresponding parts to prove ______________ are congruent.

CPCTC uses congruent triangles to prove ______________________ ____________ are congruent.

Examples: A and B are on the edges of a ravine. What is AB?

Given: YW bisects XZ, XY YZ

Prove: XYW ZYW

Given: PR bisects QPS and QRS.

Prove: PQ PS

Given: NO || MP, N P Prove: MN || OP

Given: J is the midpoint of KM and NL. Prove: KL || MN

Given: Isosceles ∆PQR, base QR, PA PB

Prove: AR BQ

Z


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T

CB

A

C

Y

X

B

A

C

A

B

D

E

Unit 1 HW: CPCTC I. Draw and label a diagram. Then solve for the variable and the missing measure or length.

1. If BATDOG, and mB=14, mG=29, and mO=10x+7 find the value of x and mO. x = ___________

mO = _________

2. If COWPIG, and CO=25, CW=18, IG=23 and PG=7x-17 find the value of x and PG.

x = ___________

PG=___________

3. If DEFPQR and DE=3x-10, QR=4x-23, and PQ=2x+7, find the value of x and EF.

x = ___________

EF = __________

II. Complete the congruence statement for each pair of congruent triangles. Then state the reason you are

able to determine the triangles are congruent. If you cannot conclude that triangles are congruent, write “none” in the blanks.

4. EFD_________________ 5. ABC_________________ 6. LKM _________________ by ________ by ________ by ________

7. ABC_________________ 8. ABC_________________

by ________ by ________

H

G

E

FD

J

MK

L


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A

B

CD

G

E

F

B

A

43

21

D

C

B

A

43

21

D

C

B

A

III. Use the given information to mark the diagram and any additional congruence you can determine from the

diagram. Then complete the triangle congruence statement and give the reason for triangle congruence. 9. 10.

Given: 1 3, 2 4 Given: ABD CBD, ADB CDB

ABC_________________ by __________ ABD_________________ by __________ 11. 12.

Given: G is the midpoint of FB and EA Given: 1 3, CDAB

ABG_________________ by __________ ABC_________________ by __________ IV. Use the given information and triangle congruence statement to complete the following.

13. ABC GEO, AB = 4, BC = 6, and AC = 8. What is the length of GO? How do you know?

14. BAD LUK, mD=52°, mB=48° and mA=80°.

a. What is the largest angle of LUK?

b. What is the smallest angle of LUK?

15. HOT SUN. SUN is isosceles. Is there enough information to determine if HOT is isosceles? Explain why

or why not.


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Cross-Sections of Solids The picture above shows the cross-section created when a knife slices an apple. A cross-section of a solid is formed when a plane passes through the solid. Below are some examples of cross-sections that can be used in various applications to explain the internal components of real-world solids. We will examine cross-sections of geometric solids. Notice the 2 cross-sections formed using a cylinder below.

If the plane is parallel to the bases, the cross-section is a circle.

If the plane passes through the cylinder vertically as shown, the cross-section is a rectangle.

Determine the shape of the cross section for each of the figures below under the given condition. Be as specific as possible. 1. Triangular Prism

a. Plane passes through the interior vertically parallel to the triangular base

b. Plane passes through the interior horizontally

2. Square Prism (not a cube and not sitting on the square base … See it pictured here -->)

a. Plane passes vertically through the center of the interior of the prism so that it is

parallel to the square base

b. Plane passes vertically through the center of the interior of the prism so that it is

not parallel to the base

c. Plane passes horizontally through the center of the interior of the prism

Source: http://images.tutorvista.com/cms/images/67/cylinder-cross-section.png

http://www.google.com/imgres?biw=1280&bih=619&tbm=isch&tbnid=8s6ghfBRlTo2IM:&imgrefurl=http://encyclopedia2.thefreedictionary.com/Automotive+Engine&docid=Es9lLy0WI8x4OM&imgurl=http://img.tfd.com/ggse/85/gsed_0001_0001_0_img0150.png&w=408&h=314&ei=Jx0JUouOKOblyQGxtYAQ&zoom=1&iact=hc&vpx=309&vpy=209&dur=6922&hovh=197&hovw=256&tx=132&ty=91&page=3&tbnh=147&tbnw=191&start=45&ndsp=28&ved=1t:429,r:47,s:0,i:226

http://www.google.com/imgres?sa=X&biw=1280&bih=619&tbm=isch&tbnid=FNPGhcqMXFdFjM:&imgrefurl=http://www.buzzle.com/articles/labeled-diagram-of-the-human-kidney.html&docid=bf3dyl2WqCREhM&imgurl=http://www.buzzle.com/images/diagrams/human-body/cross-section-of-kidney.jpg&w=500&h=500&ei=WB4JUqTBCcHXyAGqt4GoAQ&zoom=1&iact=hc&vpx=954&vpy=42&dur=3171&hovh=225&hovw=225&tx=103&ty=110&page=2&tbnh=145&tbnw=145&start=23&ndsp=32&ved=1t:429,r:29,s:0,i:174


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3. Square Pyramid

a. Plane passes horizontally through the pyramid

b. Plane passes vertically through the pyramid so that it passes through the center

c. Plane passes diagonally through the pyramid so that it is parallel to one of the sides

4. Cylinder

a. Plane passes horizontally through the cylinder

b. Plane passes diagonally through the cylinder (angled slightly from horizontal)

5. Sphere

a. Plane passes horizontally through the center

b. Plane passes horizontally through the sphere, but not through the center

c. Plane passes vertically through the center

d. Plane passes vertically through the sphere, but not through the center

e. Plane passes diagonally through the sphere

6. Which of the cross-sections in #5 above are guaranteed to be congruent?

From Core-Plus Mathematics Course I: p. 453 (Unit 6) 7.

(Additional problems can be found at http://www.glencoe.com/sites/pdfs/impact_math/ls1_c2_cross_sections.pdf)


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HW: Cross-sections (pictures are not to scale) 1) In the figure, the shaded region is a planar cross-section of the

rectangular solid. What is the area of the cross-section to the nearest inch?

2) A right circular cone with a diameter of base 8 centimeters and height 12 centimeters is shown. What is the radius of the cross-seciton that occurs 6 centimeters from the vertex, parallel to the base?

3) In the figure, the shaded areas show the intersection of two planar cross-sections that divide the large rectangular solid into four identical retangular solids. IF the four smaller rectangular solids formed by the cross-section are pulled apart, what is the total surface area of the four solids?

4) The shaded area in the figure below is a planar cross-section of the pyramid. The pyramid’sedges are all 16 centimeters long and the base of the pyramid is a square.

5) In the figure, the shaded area is a planar cross-section that is perpendicular to the bases of the right cylinder. What is the area of the cross-section?


30

Partitioning a Line Segment

Example 1) Segment AB has coordinates A(-7,4) and B(8,9). Find the coordinates of a point C that divides

segment AB in a ratio of 3:2.

a. Draw 𝐴𝐵 with point 𝐶 between points 𝐴 and 𝐵. Label the coordinates of 𝐴, 𝐵, and 𝐶, and label the lengths

of AC and CB.

b. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑥 direction.

c. Solve your proportion to find the 𝑥 coordinate of 𝐶.

d. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑦 direction.

e. Solve your proportion to find the 𝑦 coordinate of 𝐶.

f. Write 𝐶 as an ordered pair.

g. Check your answer by using the distance formula to verify that AC:CB is 3:2.


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Example 2) Find point C on 𝐴𝐵 such that 𝐴𝐶: 𝐶𝐵 = 2: 1, when 𝐴 = (1, 1) and 𝐵 = (4, 1).

a. Draw 𝐴𝐵 with 𝐶 between 𝐴 and 𝐵. Label the coordinates of 𝐴, 𝐵, and 𝐶, and label the lengths of AB and BC.

b. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑥 direction.

c. Solve your proportion to find the 𝑥 coordinate of 𝐶.

d. Set up a proportion describing the lengths of 𝐴𝐶 and 𝐶𝐵 in the 𝑦 direction.

e. Solve your proportion to find the 𝑦 coordinate of 𝐶.

f. Write 𝐶 as an ordered pair.

g. Check your answer by using the distance formula to verify that 𝐴𝐶 is twice as long as 𝐶𝐵.


32

HW: Partitioning a Line Segment


33


34

Unit Review

1. Given BED BUG, find x and BU if BE = 6x+16, UG=10x-2 and ED = 6x+6.

x=__________ BU=________________

2. In ∆ABC, Ab, AC = 6x-5, BC = 3x+13, and AB = 4x+7. Find x and the length of the base.

x ___________

base = ___________

3. In an isosceles triangle, a vertex angle measures 36⁰. What is the measure of each base angle?

____________ 4. Solve for the value of x in parts a and b. Solve for the measure of angle 1 in part c. a. b. c. 5. Decide whether it is possible to prove the triangles are congruent. If yes, then state which congruence

postulate you would use.

a. b. c. d.

6. If ABCPQR and C=2x+2, R=3x-18, find the value of x.

7. ∆XYZ ∆JKL. IfY=14-x, K=2x+50 and L=-4x find the Zm .


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8. IfABCLMN, and AB=4x-y, LM =2x-2y, BC = x-37 and MN =21 find the value of x and y. 9. ∆PQR ∆XYZ. List three pairs of congruent angles.

____________________ _____________________ ____________________ 10. Suppose ∆ABC ∆EFG. For each of the following, name the corresponding part.

A : ______________ BCA : ______________ AC : _______________

11. In ∆EFG, E = 6x – 8, F = 7x + 3, and G = 3x – 7. Find the measure of each angle.

12. Two angles of a triangle measure 18 o and 77 o. Find the measure of the third angle.

A. 19o B. 36 o C. 59 o D. 64 o E. 85 o

13.

A. PMN

B. NPM

C. NMP

D. MNP

14. In the given triangles, ABCXYZ Which two statements identify corresponding congruent parts for these triangles?

A. AB XY and CY

B. AB YZ and CX

C. BC XY and AY

D. BC YZ and AX

15. BOY DEA. YB = 4x+8, AD = 60, AE = 80, AND DE = 40. Find x.

16. EXC STP. C = 3x+12, T = 81, and X = 12x – 15 . Find m C .


36

50

75

40

C

A

B

Y

Z

67.545

B

A

C Y

X

Z

12

18

9

6

C

B

A

E

D

105

3 6F

G

R

N

16

128

10

15

20

A

B C M

U

K

17. MNODEF. Calculate the value for x, y, and z. 𝑚∠𝑀 = 50 x = ________ 𝑚∠𝑁 = 60 𝑚∠𝑂 = 70 y = ________ 𝑚∠𝐷 = (2𝑥 − 20)

𝑚∠𝐸 = (1

2𝑦 + 10) z = ________

𝑚∠𝐹 = (10 + 𝑧)

18. The figure shows . ABCEDC. C is the midpoint of BD and AE. What reason would you use to prove the triangles are congruent, if any?

If the triangles are congruent, complete the statement: A _____

19. If ∆JKM ∆RST, how do you know JK RS? A. Definition of a line segment C. SSS Postulate B. CPCTC D. SAS Postulate

20. ΔWHY ~ ΔGEO. mW = 55, mE = 60. Find mY. 21. ΔMAD ~ ΔCOW. MA = 8, AD = 5, CO = 6. Find the scale factor of ΔMAD to ΔCOW and the length of OW.

Scale factor = ________ OW = _________

22. If two triangles are congruent, what is the scale factor? Scale factor = ____ : ____ Determine whether the following triangles are similar. If so, complete the similarity statement and a state a specific reason. If not, write “not similar”. (Figures are not drawn to scale!) 23. 24. 25. 26. 27. 28.


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Homework Answers:

Similar Triangles: Page 1) Not similar

2) Similar, SSS, FGH

3) Similar, SAS, VLM

4) Similar, AA, TUV 5) Not similar

6) Not similar

7) Similar SSS, QRS 8) Not similar

9) Similar, AA, HTS

10) Similar, SAS, UVW

11) Similar, SSS, FRS

12) Similar, SAS, CDE 13) 22 14) 54 15) 9

16) 11 17) 8 18) 9 19) 7 20) 11

Congruency and Rigid Motion: Pages

1 a) Translation: (x, y) (x + 6, y – 7) b) A translation is a rigid motion which preserves

length, thus creating a figure with the same size and shape as the pre-image.

2 b) MA = M’A’ = √20

AP = A’P’ = √17

MP = M’P’ = √45 c) A counterclockwise rotation of 180° 3 b) Reflection across the line y = x 4 b) Yes. A reflection is a rigid motion which

preserves length, thus creating a figure with the same size and shape as the pre-image.

d) Yes. A translation is a rigid motion which preserves length, thus creating a figure with the same size and shape as the pre-image.

e) Yes. Both triangles are congruent to RST, therefore by the transitive property of congruence, they are congruent to each other. Or rigid motion congruence is transitive.

5 a) Counterclockwise rotation of 90o around the origin. b) Yes. A rotation is a rigid motion which preserves

length, thus creating a figure with the same size and shape as the pre-image.

6 b) A dilation centered at (0,0) with a scale factor 2. c) While these triangles are the same shape, they

are not the same size. A dilation is NOT a rigid motion and does not preserve the congruency of figures. These triangles are similar, but not congruent.

SSS, SAS, ASA, AAS, and HL Congruence: Pages 1) Not congruent 2) HL 3) Not congruent 4) SAS 5) SSS 6) SAS

7) SAS 8) AAS 9) HL 10) ASA 11) ASA 12) ASA

13) SSS 14) Not congruent 15) AAS 16) SAS

17) I C

18) SQR CRQ

19) UVW EWV or

VWU WVE

20) 𝑌𝑋̅̅ ̅̅ 𝑌𝐾̅̅ ̅̅ 21) G T

22) 𝐸𝐶̅̅̅̅ 𝑁𝐿̅̅ ̅̅

Triangle Sum Theorem, Isosceles Triangle Theorem & Converse, Midsegments: Pages 1) x = 8 y = 10 z =

10 2) x = 6.5 3) x = 20 4) x = 9 5) x = 31 6) x = 10

7) x = 35/4 y = 15 8) x = 3 y = 7 9) 36.5 10) 57.275 11) 80 12) 45 13) 53

14) 11 15) 5 16) 6 17) 41 18) 8 19) 6 20) 5

21) 55 22) 8.63

23) a D A_.

b. ECB EBC. c. 45°.


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CPCTC and Naming Congruent Triangles: Pages 1) x = 13

𝑚∠𝑂= 137 deg 2) x = 5

PG= 18 3) x = 17

EF = 45

4) GHD; SSS

5) TBC; ASA

6) JKM; SSS, ASA or SAS

7) not congruent

8) EDC; AAS

9) DCB; ASA

10) CBD; ASA

11) EFG; SAS

12) DCB; SAS 13) GO = 8

corresponds to AC

14) a) U

b) L

15) Yes, they are congruent, so

HOT has the exact same size and shape and

SUN.

Cross-sections: Page 1) 3,225 in2 2) 2 cm 3) 1,536 cm2 4) 54.7 cm2 5) 720 cm2 Partitioning a Line Segment: Pages 1) (1, 5) 2) (1, 2)

3) (1

5, −

8

5)

4) (−1, −1

3)

(1,4

3)

5) (13, 12)

6) (7,5) 7) (4,10) 8) (3,13) 9) (7,11)

Test Review: Pages 1) x=2, BU=28 2) x=6, base=31 3) 72 4) a) -15

b) 1 c) 74°

5) a) SSS b) Not c) SSS d) SAS

6) 20 7) -48 8) x=3, y=-6

9) PX

QY

RZ 10) E, FGE, EG 11) 64, 87, 29 12) E

13) D 14) D 15) 13 16) 36 17) x=35

y=100 z=60

18) SAS, E 19) B 20) 65

21) 4:3 15/4

22) 1:1 23) PPP 24) AA 25) Not 26) PAP 27) Not 28) PAP


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unit 1b modeling with geometry:...

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