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Foundations of Math II
Unit 3: Similarity and
Congruence
Academics
High School Mathematics
1
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
19
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
35
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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