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Foundations of Math II

Unit 3: Similarity and Congruence

Academics

High School Mathematics

3.1 Warm Up

1. Jill and Bill are doing some exercises. Jayne Funda, their instructor, gently implores “Touch your nose to your knees, maggots!” Their attempts to please Ms. Funda are shown below. Bills says, “I’m doing better than you, Jill. My nose is much closer to my knees!”Jill replies, “That isn’t a fair comparison, Bill.”With whom do you agree? Who is doing a better job? Explain your answer.

2. The perimeter of COW is 12 units.

a) Find possible lengths for CO, OW , and CW .

b) Find four more sets of possible lengths.

c) How many answers are possible?

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

1

Jill Bill

3.2 Warm Up

1. Which of the figures below could be the image of figure a when dilated? Explain why or why not for each figure.

2) a) Draw a line that passes through the origin of a coordinate plane and forms a 45 angle with the x-axis.

b) Find the coordinates of at least three points on the line. c) Write an equation for the line. What do you notice?


3.2 Practice with Dilations on the Coordinate Plane

Graph three points that lie in three different quadrants and connect them to form a triangle. Label the vertices of the triangle as TRI.

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a

s

g

r

p

e

f

c

Record the coordinates of the triangle in the table below. Then find and apply the algebraic rules for each of the scale factors listed below. Graph and label each image.

Scale Factor32 2

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Algebraic Rule (x, y) (x, y) (x, y)

T ( , ) T’ ( , ) T’’ ( , ) T’’’ ( , )

R ( , ) R’ ( , ) R’’ ( , ) R’’’ ( , )

I ( , ) I’ ( , ) I’’ ( , ) I’’’ ( , )

What would each scale factor be if written as a percent?

7

8

Explain why or why not for each pair.

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Find the scale factor. The pre-image is indicated by an arrow.

3.3 Warm Up

1. Draw each of the following dilations of quadrilateral BRIA:a. 150% scale factor using center X.b. 32 scale factor using center Y.c. 1.5 scale factor using center I.d. What do you notice?


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Y

A

I

B

R

X

O

C

NP

AR

3.4 Warm Up1. RAP is an image of CON using a dilation. Find point Z, the center of dilation, and also

the scale factor.

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii3.4 Investigation: Geometric Mean

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1. Use your protractor to find the measure of the following angles.

mABC = ___________ mBCD = __________ mCAB = __________

mBDC = __________ mADC = _________

mBCD= __________ mDCA= _________

What type of segment is CD called in ∆ABC?

What kind of triangles are ABC, ADC, and BDC?

1. Trace ∆ ABC with a ruler on patty paper and carefully cut out ∆ADC and ∆BDC. You will need to place the letter of each vertex in the interior of each triangle on the patty paper so that you can still tell what ∆ you are working with after you cut it out (see below).

2. Next trace and cut out ∆ ABC. Be sure to label each vertex in the interior of the triangle.

3. Stack the cut out triangles so that that corresponding sides match up. Note that the three right triangles are similar to each other. Write a similarity statement for the three triangles. ABC _______________________________

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We know that the ratios of corresponding sides of similar triangles are proportional,4. Using ∆BDC and CDA fill in the proportion: BDDC=¿ So DC must be the geometric mean of ___________ and ___________.5. Using ∆BDC and ∆ABC fill in the proportion: ABBC=¿ So BC must be the geometric mean of ___________ and ___________.6. Using ∆ADC and ∆ABC, fill in the proportion: ABAC=¿ So AC must be the geometric mean of ___________ and ___________.

a b

n

h

m

c7. Write three proportions for the picture above using what you learned from this activity.

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Geometric Mean: Example Problems

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1. 2.

3. 4.

3.4 Geometric Mean Practice

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3.5 Warm Up

1) a) If a line has a slope greater than 1, what angle might it make with the x-axis?

b) If a line has a slope less than 1, what angle might it make with the x-axis?

c) If a line has a slope equal to 1, what angle might it make with the x-axis?

Adapted from Geometry: A Moving Experience developed by the Curriculum Research & Development Group, College of Education at the University of Hawaii3.5 Midsegment Example ProblemsExample 1

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Find x.

Example 2DE is the midsegment of ABC. Find x, AC, and ED.

Example 3MN is the midsegment of JKL.MN = 2x + 1KJ = 5x – 8 Find x, MN, and KJ.

Example 4

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287x

3.5 Midsegments – Show What You Know!1) XY is the midsegment of RST. Find each requested measure based on the given information.

a) XY = 16, RS = ?

b) RS = 22, XY = ?

c) XY = 5x, RS = 15, x = ?

d) mR = 23, mTXY = ?

e) mXYS = 137, mYSR

2) Find x and y.

3) Find MS, PT, and ST.

4)a)

b)

c)

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3y

18

3.6 Warm Up

1) a) A line forms an angle measuring less than 45 with the x-axis. What might its slope be?

b) A line forms an angle measuring more than 45, but less than 90, with the x-axis. What might its slope be?

c) What might the slope be if the line forms an obtuse angle with the x-axis?


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3.7 Warm Up

1) A line passes through the origin and the point A(7, 3). Without graphing the line, what can you conclude about the angle it will form with the x-axis?


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3.9 Flow Proof Examples

1)

2)

3)

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C is the midpoint of

ADA D

1 2

ABC ___________

AC DC

MN ∥PT

NO ¿

MNO ___________

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34

BD bisects ABC

AB CB

ABD ___________

ABD CBD

BD BD

ABD ___________

Given

Definition of Angle Bisector

4)

5)

6)

40

AS bisects MP

MAS ___________

Given

Reflexive Property of Congruence

E is the midpoint of

ADA D

B C

ABE ___________

AE DE

C is the midpoint of C is the midpoint of

3.10 Warm-up

1. Erica builds a ramp that makes a 45° with the ground. Her support board is 10 feet from the beginning of the ramp.

a. How high is her support board?b. How long is her ramp?

2. Line m forms a 40° angle with the x-axis. Find the slope of line m. Explain your answer?


3.10 Practice1)

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2)

3)

3.11 Warm Up

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1 and 2 are right angles

AD CB

ABD CDBAll right angles are congruent

ABD CBD

AB _________

Converse of the Isosceles Triangle

Theorem

SS

A

1) Bill builds a ramp at a 56 angle with the ground. He uses a 12-foot support board and finds that the support board must be 8 feet from the beginning of the ramp in order to make the 56 angle. Jill also builds a ramp at a 56 angle with the ground. She uses a 9-foot support board. How far should her board be from the beginning of her ramp? Illustrate and explain your answer.


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Vocabulary Word

Definition CharacteristicsPicture and/or

SymbolReal Life Examples

AAS

ASA

congruent

dilation

44

Vocabulary Word



extremes

flow proof

geometric mean

means

45

Vocabulary Word



midsegment

Midsegment Theorem

proof

proportion

46

Vocabulary Word



SAS

scale factor

side

similar

47

Vocabulary Word



SSS

triangle

Triangle Angle Sum Theorem

vertex

48

Vocabulary Word



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