Foundations of Math II

Unit 3: Similarity and

Congruence

Academics

High School Mathematics

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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 𝐶𝑂̅̅ ̅̅ , 𝑂𝑊̅̅ ̅̅ ̅, and 𝐶𝑊̅̅ ̅̅ ̅.

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

Jill Bill

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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?

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

a

s

g

r

p

e

f

c

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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.

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 Factor 𝟑

𝟐 2

𝟏

𝟐

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?

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Explain why or why not for each pair.

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

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3.3 Warm Up

1. Draw each of the following dilations of quadrilateral BRIA:

a. 150% scale factor using center X.

b. 3

2 scale factor using center Y.

c. 1.5 scale factor using center I.

d. What do you notice?

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

Y

A

I

B

R

X

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3.4 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 Hawaii

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3.4 Midsegment Example Problems

Example 1

Find x.

Example 2

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

Example 3

MN is the midsegment of JKL.

MN = 2x + 1

KJ = 5x – 8

Find x, MN, and KJ.

Example 4

28 7x

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3.4 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)

3y

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3.5 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?

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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3.6 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?

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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3.7 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?

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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3.8 Warm Up

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.

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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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

AAS

ASA

congruent

dilation

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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

extremes

flow proof

geometric mean

means

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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

midsegment

Midsegment Theorem

proof

proportion

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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

SAS

scale factor

side

similar

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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

SSS

triangle

Triangle Angle Sum Theorem

vertex

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Vocabulary Word

Definition Characteristics Picture and/or

Symbol Real Life Examples

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