Transcript Similarity - MHSmathONLINE

Warm-Up
Since they are polygons, what two things
must be true about triangles if they are
similar?
Similar Polygons
Two polygons are similar polygons iff the
corresponding angles are congruent and the
corresponding sides are proportional.
Similarity Statement:
N
C
M
CORN ~ MAIZ
Corresponding Angles:
C  M
C
R  I
O  NA
N  Z
O
R
A
M
StatementOof Proportionality:
R
CO  OR  RN  NC
MA AI IZ ZM
A
I
Z
Example 1
Triangles ABC
and ADE are
similar. Find the
value of x.
D
B
A
6 cm
9 cm
8 cm
C
x
E
Example 2
Are the triangles below similar?
8
4
6
3
37
53
5
10
Do you really have to check all the sides and angles?
6.4-6.5: Similarity Shortcuts
Objectives:
1. To find missing measures in similar
polygons
2. To discover shortcuts for determining that
two triangles are similar
Investigation 1
In this Investigation we will check the first
similarity shortcut. If the angles in two
triangles are congruent, are the triangles
necessarily similar?
F
C
A
50
40
B
D
50
40
E
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
equal sensible values of your choosing.
C
A
50
40
B
Investigation 1
Step 1: Draw ΔABC where m<A and m<B
equal sensible values of your choosing.
Step 2: Draw ΔDEF where m<D = m<A and
m<E = m<B and AB ≠ DE.
F
C
A
50
40
B
D
50
40
E
Investigation 1
Now, are your triangles similar? What would
you have to check to determine if they are
similar?
F
C
A
50
40
B
D
50
40
E
Angle-Angle Similarity
AA Similarity
Postulate
If two angles of one
triangle are
congruent to two
angles of another
triangle, then the two
triangles are similar.
Example 3
Determine whether the triangles are similar.
Write a similarity statement for each set of
similar figures.
Thales
The Greek mathematician
Thales was the first to
measure the height of a
pyramid by using
geometry. He showed
that the ratio of a
pyramid to a staff was
equal to the ratio of one
shadow to another.
Example 4
If the shadow of the pyramid is 576 feet, the
shadow of the staff is 6 feet, and the height
of the staff is 5 feet, find the height of the
pyramid.
Example 5
Explain why Thales’ method worked to find
the height of the pyramid?
Example 6
If a person 5 feet tall casts a 6-foot shadow
at the same time that a lamppost casts an
18-foot shadow, what is the height of the
lamppost?
Investigation 2
What if you decide to
indirectly measure a
height on a day when
there are no shadows?
The following GSP
Animation will help you
discover an alternate
method of indirect
measurement using a
mirror.
Example 7
Your eye is 168
centimeters from
the ground and you
are 114 centimeters
from the mirror.
The mirror is 570
centimeters from
the flagpole. How
tall is the flagpole?
Investigation 3
Each group will be given one
of the three candidates for
similarity shortcuts. Each
group member should start
with a different triangle and
complete the steps outlined
for the investigation.
Share your results and
make a conjecture based
on your findings.
Side-Side-Side Similarity
SSS Similarity Theorem:
If the corresponding side lengths of two
triangles are proportional, then the two
triangles are similar.
Side-Angle-Side Similarity
SAS Similarity Theorem:
If two sides of one triangle are proportional to two
sides of another triangle and the included angles
are congruent, then the two triangles are similar.
Example 8
Are the triangles below similar? Why or why
not?
Example 9
Use your new conjectures to find the missing
measure.
28
24
24
x
18
y
Example 10
Find the value of x that makes ΔABC ~
ΔDEF.
Assignment
• P. 384-387: 1-4, 7,
8, 10, 12, 14-17,
20, 30, 31, 32, 36,
41, 42
• P. 391-395: 4, 6-8,
10-14, 33, 39, 40
• Challenge
Problems