Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 9: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria
Learning Target I can use the side-angle-side and side-side-side criterion for two triangles to be similar to
solve triangle problems.
Concepts to remember from Lesson 8
Two triangles △ 𝐾𝑀𝐿 and △ 𝐻𝐽𝐼 are similar if there is a similarity
transformation that maps △ 𝐾𝑀𝐿 and △ 𝐻𝐽𝐼 .
So △ 𝐾𝑀𝐿~ △ 𝐻𝐽𝐼, the similarity transformation takes 𝐾 to 𝐻, 𝑀 to 𝐽,
and 𝐿 to 𝐼, such that the corresponding angles are equal in measurement
and the corresponding lengths of sides are proportional.
Also we learned AA similarity criteria - Two triangles can be considered
similar if they have two pairs of corresponding equal angles.
A New Condition for Similarity: S-A-S Similarity
Two triangles are ______________________ if they have one pair of _____________
____________that are congruent and the sides adjacent to that angle are proportional
This is called the ____________ ______ _____________ criterion.
How do we prove that two tringles are similar?
Given two triangles ∆𝐴𝐵𝐶 and∆𝐴’𝐵’𝐶’ so that
𝐴′ 𝐵′
𝐴𝐵
=
𝐴′ 𝐶 ′
𝐴𝐶
and
𝑚∠𝐴 = 𝑚∠𝐴′ , then the triangles are similar, ∆𝐴𝐵𝐶 ~∆𝐴′ 𝐵 ′ 𝐶 ′ .
=
=
and 𝑚∠
= 𝑚∠
, then
Example 1. Using the definition above, are the two triangles below similar? Explain your answer
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Example 2. Use the figure at right to answer the following questions.
a) Name and state all the triangles that you see in Figure 1. Draw and
label each triangle separately in the space below.
b) Are the corresponding sides of these two triangles proportional? Show the work that leads to your answer.
c) Are the included angles of these two triangles congruent? Give a reason for your answer.
d) Are the two triangles similar? If so, write a similarity statement.
Example 3. Examine the figure, and answer the questions to determine whether or not the triangles
shown are similar.
a. What information is given about the triangles in Figure 3?
b. How can the information provided be used to determine
whether △ 𝑨𝑩𝑪 is similar to △ 𝑨′𝑩′ 𝑪′?
c. Compare the corresponding side lengths of △ 𝑨𝑩𝑪 and △ 𝑨′𝑩′ 𝑪′. What do you notice?
d. Based on your work in parts (a)–(c), make a conjecture about the relationship between △ 𝑨𝑩𝑪 and △ 𝑨′𝑩′ 𝑪′. If
so write a similarity statement.
_______________________________________________________________________________________________
_______________________________________________________________________________________________
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Another condition for Similarity is Side-Side-Side (SSS)
If all three sides of one triangle are _____________________________ all three sides of another triangle,
then the two triangles are _____________________.
This is called _______ _______ _______ ______________________.
Example 4
1. What information is given about the triangles in
Figure 4?
2. How can the information provided be used to
determine whether∆𝐴𝐵𝐶 is similar to ∆𝐴′𝐵 ′ 𝐶 ′ ?
3. Compare the corresponding side length of
∆𝑨𝑩𝑪 and ∆𝑨′𝑩′ 𝑪′ . What do you notice?
4. Based on your work in parts (a)–(c), make a conjecture about the relationship between ∆𝐴𝐵𝐶 and
∆𝐴′𝐵 ′ 𝐶 ′ . If so write a similarity statement.
_______________________________________________________________________________________
_______________________________________________________________________________________
What are the three conditions we can use to prove that triangles are similar?
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
Lesson 9
M2
Period: ________ Date: __________
Lesson 9: The Side-Angle-Side (SAS) and Side-Side-Side (SSS) Criteria
Problem Set
1. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
2. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
3. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
𝟓 𝟑
≠
𝟑 𝟏
There is no information about the angle measures other
than the right angle, so we cannot use AA to conclude the
triangles are similar. We only have information about two of
the three side lengths for each triangle, so we cannot use SSS
to conclude they are similar. If the triangles are similar, we
would have to use the SAS criterion, and since the side lengths are not proportional, the triangles
shown are not similar.
Lesson 9
NYS COMMON CORE MATHEMATICS CURRICULUM
M2
Name:__________________________________
Period: ________ Date: __________
4. Given ∆𝐴𝐵𝐶 and ∆𝐿𝑀𝑁 in the diagram below, and ∠𝐵 ≅ ∠𝐿,
determine if the triangles are similar. If so, write a similarity
statement, and state the criterion used to support your claim.
5. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters.
b.
.
a.
𝑚 =___________________
𝑛 = _____________
𝑠 =___________________
𝑡 = _____________
6. One triangle has a 𝟏𝟐𝟎° angle, and a second triangle has a 𝟔𝟓° angle. Is it possible that the two triangles
are similar? Explain why or why not.
7. A right triangle has a leg that is 𝟏𝟐 𝐜𝐦 long, and another right triangle has a leg that is 𝟔 𝐜𝐦 long. Are the
two triangles similar or not? If so, explain why. If not, what other information would be needed to show
they are similar?
The two triangles may or may not be similar. There is not enough information to make this claim. If the second leg of the first triangle is
twice the length of the second leg