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SUBJECT
Geometry
UNIT TITLE Similarity
Topic: Proving Triangles Similar (AA, SAS, and SSS)
Objective:
Students will be able to identify and use AA, SAS, and SSS similarity
to solve a variety of problems including real world applications.
Core Content for Assessment:
MA-11-3.1.6 Students will apply the concepts of congruence and
similarity to solve real world problems.
Materials/Resources:
Discovery Activity (included in this folder)
Notes (included in this folder)
Possible Test Questions (included in this folder)
Warm Up/Bell Ringer/Do Now:
Included in this folder
Procedures:
Warmup
Investigation
Notes
Review and assign homework accordingly
(this is very generic, modify to fit your usual classroom procedure)
Lesson Assessments:
Included in this folder
Bellringer for Proving Triangles Similar

What does it mean for two triangles to be congruent?

You have already worked with proving triangles congruent in several
situations (SAS, SSS, ASA, AAS).

What does the A stand for and what does the S stand for?
A is __________________ and S is __________________

Today we will be working with proving triangles similar, not proving
triangles congruent.

What does it mean for two triangles to be similar?

What is the difference between proving triangles congruent and
proving triangles similar?

If triangles are congruent, does that mean that they are similar?
Explain.

If two triangles are similar, do they have to be congruent? Explain.

Today we will be working with proving triangles similar using AA, SAS,
and SSS. What does each one of these stand for?
Investigation: Triangles with Two Pairs of Congruent Angles
 Draw and label 2 triangles of different sizes, each with a
50◦ and a 60◦ angle (using your protractor).
 Measure the sides of each triangle to the nearest
millimeter, using your ruler.
 Find the ratio of lengths of each corresponding side and
record this information.
1) What conclusions can you make about the two
triangles?
2) Complete this conjecture:
If two angles of one triangle are congruent to two
corresponding angles of another triangle, then the
triangles are ____________________.
Notes: Proving Triangles Similar Using AA, SAS, and SSS
From our investigation we were able to notice that:
Postulate: Angle-Angle Similarity Postulate
If two angles of one triangle are congruent to two angles of another
triangle, then the triangles are similar.
D
G
28°
28°
83°
83°
C
F
E
H
Write a similarity statement for these two triangles.
Why does this similarity make sense?
Example 1: Are the following triangles similar? If so why? If not, why not?
A
C
D
45
45
B
F
Theorem: Side-Angle-Side Similarity (SAS) Theorem
If an angle of one triangle is congruent to an angle of a second triangle, and
the sides including the two angles are proportional, then the triangles are
similar.
B
D
A
C
Write a similarity statement for the two triangles above with the correct
reasoning.
Example 2: Are the following triangles similar? If so why? If not, why not?
A
12
6 C
B
8
4
D
E
Theorem: Side-Side-Side Similarity (SAS) Theorem
If the corresponding sides of two triangles are proportional, then the
triangles are similar.
A
R
16
15
8
B
6
C T
7.5
3
S
Write a similarity statement for these two triangles.
Example 3: Are the following triangles similar? If so why? If not, why not?
A
R
6
6
4
B
C T
4
4
S
3
Indirect Measurement
You can often use similar triangles to measure quantities in the real world
that would be extremely difficult and time consuming to measure directly.
This is called indirect measurement.
Here is a real world example of using indirect measurement.
Example 4:
Campsites F and G are on opposite sides of a lake. A survey crew made the
measurements shown on the diagram. What is the distance between the two
campsites? The diagram is not to scale.
F
82 m
82.8 m
~~
~
~
~~
~
~
~~
47 m
92 m
~~
73.8 m
G