Download Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar

Lesson 17A
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar
Learning Target
 I can use the side-angle-side criterion for two triangles to be similar to solve triangle problems.
Opening exercise
State the coordinates of the image of the following composition of transformations.
𝐷2 ∘ 𝑟𝑦−𝑎𝑥𝑖𝑠 ∘ 𝑅90° (∆𝐷𝑅𝐺)
Original
coordinates
𝐷(
,
)
𝑅(
,
)
𝐺(
,
)
Is this composition a similarity transformation?
Lesson 17A
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Concepts to remember from lesson 14-15
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?
𝐴′ 𝐵′
𝐴′ 𝐶 ′
Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴’𝐵’𝐶’ so that
=
and
𝐴𝐵
𝐴𝐶
′
′ ′ ′
𝑚∠𝐴 = 𝑚∠𝐴 , then the triangles are similar, △ 𝐴𝐵𝐶 ~ △ 𝐴 𝐵 𝐶 .
=
=
and 𝑚∠
= 𝑚∠
, than
Example 1) Using the definition above, are the two triangles below similar? Explain your answer
Lesson 17A
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.
1. What information is given about the triangles in Figure 2?
Ans: We are given that ∠𝑷 is common to both triangle △ 𝑷𝑸𝑹 and triangle
△ 𝑷𝑸′𝑹′. We are also given information about some of the side lengths.
2. How can the information provided be used to determine whether △ 𝑷𝑸𝑹
is similar to △ 𝑷𝑸′𝑹′?
Ans: We know that similar triangles will have ratios of corresponding sides that are proportional; therefore, we can use the
side lengths to check for proportionality.
3. Compare the corresponding side lengths of △ 𝑷𝑸𝑹 and △ 𝑷𝑸′𝑹′. What do you notice?
𝟑
𝟏𝟑
≠
𝟐𝟕 ≠ 𝟐𝟔
𝟐
𝟗
The side lengths are not proportional.
4. Based on your work in parts (a)–(c), draw a conclusion about the relationship between △ 𝑷𝑸𝑹 and
△ 𝑷𝑸′𝑹′. Explain your reasoning.
Lesson 17A
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar
Classwork
1. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
3.13
3
=
1.565 1.5
Yes, the triangles shown are similar. △ 𝐴𝐵𝐶 ~ △ 𝐷𝐸𝐹 by SAS because m∠𝐵 = m∠𝐸, and the adjacent sides are proportional.
2. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
Yes, the triangles shown are similar. △ 𝑨𝑩𝑪 ~ △ 𝑨𝑫𝑬 by AA because
𝐦∠𝑨𝑫𝑬 = 𝐦∠𝑨𝑩𝑪, and both triangles share ∠𝑨.
.
3. Are the triangles shown below similar? Explain. If the triangles are similar, write the similarity statement.
𝟓
𝟑
≠𝟏
𝟑
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 17A
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
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.
In comparing the ratios of sides between figures, I found that
the cross products of the proportion
𝟖
𝟑
=
𝟐
𝟏𝟐
𝟑
𝟑
𝟒
𝟒
𝑨𝑩
𝑴𝑳
=
𝑪𝑩
𝑳𝑵
because
are both 𝟑𝟖. We are given that
∠𝑳 ≅ ∠𝑩. Therefore, ∆𝑨𝑩𝑪~∆𝑴𝑳𝑵 by the 𝑺𝑨𝑺 criterion for proving similar
triangles.
5. For each pair of similar triangles below, determine the unknown lengths of the sides labeled with letters.
a.
b.
.
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.
No, the triangles cannot be similar because in the first triangle, the sum of the remaining angles is 𝟔𝟎°, which means that it is not possible
for the triangle to have a 𝟔𝟓° angle. For the triangles to be similar, both triangles would have to have angles measuring 𝟏𝟐𝟎° and 𝟔𝟓°, but
this is impossible due to the angle sum of a triangle.
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 of the first triangle, then the triangles are similar by SAS criterion for showing similar triangles.
Lesson 17A
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:__________________________________
M2
Period: ________ Date: __________
Lesson 17A: The Side-Angle-Side (SAS) Two Triangles to be Similar
Homework - Problem Set
1. For each part (a) through (d) below, state which of the three triangles, if any, are similar and why.
Triangles 𝑩 and 𝑫 are the only similar triangles because they have the same angle measures. Using the angle sum of a triangle, each of the
triangles 𝑩 and 𝑫 have angles of 𝟕𝟓°, 𝟔𝟎°, and 𝟒𝟓°.
2. For each part (a) through (c) below, state which of the three triangles, if any, are similar and why.
Triangles 𝑨 and 𝑩 are similar because they have two pairs of corresponding sides that are in the same ratio and their included angles are
equal measures. Triangle 𝑪 cannot be shown similar because even though it has two sides that are the same length as two sides of triangle
𝑨, the 𝟕𝟎° angle in triangle 𝑪 is not the included angle and, therefore, does not correspond to the 𝟕𝟎° angle in triangle 𝑨.
3. For each given pair of triangles, determine if the triangles are similar or not, and provide your
reasoning. If the triangles are similar, write a similarity
statement relating the triangles.
The triangles are similar because, using the angle sum of a triangle, each triangle
has angle measures of 𝟓𝟎°, 𝟔𝟎°, and 𝟕𝟎°. Therefore, ∆𝑨𝑩𝑪~∆𝑻𝑺𝑹.