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AA Similarity
Bill Zahner
Lori Jordan
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Printed: February 26, 2016
AUTHORS
Bill Zahner
Lori Jordan
www.ck12.org
C HAPTER
Chapter 1. AA Similarity
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AA Similarity
Here you’ll learn how to determine whether or not two triangles are similar using AA Similarity.
What if you were given a pair of triangles and the angle measures for two of their angles? How could you use this
information to determine if the two triangles are similar? After completing this Concept, you’ll be able to use AA
Similarity to decide if two triangles are similar.
Watch This
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/137606
CK-12 Foundation: Chapter7AASimilarityA
Watch this video beginning at the 2:09 mark.
MEDIA
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URL: http://www.ck12.org/flx/render/embeddedobject/1302
James Sousa: Similar Triangles
MEDIA
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URL: http://www.ck12.org/flx/render/embeddedobject/1357
James Sousa: Similar Triangles by AA
Guidance
The Third Angle Theorem states if two angles are congruent to two angles in another triangle, the third angles
are congruent too. Because a triangle has 180◦ , the third angle in any triangle is 180◦ minus the other two angle
measures. Let’s investigate what happens when two different triangles have the same angle measures.
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Investigation: Constructing Similar Triangles
Tools Needed: pencil, paper, protractor, ruler
1. Draw a 45◦ angle. Extend the horizontal side and then draw a 60◦ angle on the other side of this side. Extend
the other side of the 45◦ angle and the 60◦ angle so that they intersect to form a triangle. What is the measure
of the third angle? Measure the length of each side.
2. Repeat Step 1 and make the horizontal side between the 45◦ and 60◦ angle at least 1 inch longer than in Step
1. This will make the entire triangle larger. Find the measure of the third angle and measure the length of each
side.
3. Find the ratio of the sides. Put the sides opposite the 45◦ angles over each other, the sides opposite the 60◦
angles over each other, and the sides opposite the third angles over each other. What happens?
AA Similarity Postulate: If two angles in one triangle are congruent to two angles in another triangle, the two
triangles are similar.
The AA Similarity Postulate is a shortcut for showing that two triangles are similar. If you know that two angles in
one triangle are congruent to two angles in another, which is now enough information to show that the two triangles
are similar. Then, you can use the similarity to find the lengths of the sides.
Example A
Determine if the following two triangles are similar. If so, write the similarity statement.
Find the measure of the third angle in each triangle. m6 G = 48◦ and m6 M = 30◦ by the Triangle Sum Theorem.
Therefore, all three angles are congruent, so the two triangles are similar. 4FEG ∼ 4MLN.
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Chapter 1. AA Similarity
Example B
Determine if the following two triangles are similar. If so, write the similarity statement.
m6 C = 39◦ and m6 F = 59◦ . The angles are not equal, 4ABC and 4DEF are not similar.
Example C
Are the following triangles similar? If so, write the similarity statement.
Because AE || CD, 6 A ∼
= 6 D and 6 C ∼
= 6 E by the Alternate Interior Angles Theorem. Therefore, by the AA Similarity
Postulate, 4ABE ∼ 4DBC.
Watch this video for help with the Examples above.
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/137607
CK-12 Foundation: Chapter7AASimilarityB
Guided Practice
Are the following triangles similar? If so, write a similarity statement.
1.
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2.
3.
Answers:
1. Yes, 4DGE ∼ 4FGD ∼ 4FDE.
2. Yes, 4HLI ∼ 4HMJ.
3. No, though 6 MNQ ∼
= 6 ONP because they are vertical angles, we need to have two pairs of congruent angles in
order to be able to say that the triangles are similar.
Explore More
Use the diagram to complete each statement.
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Chapter 1. AA Similarity
1. 4SAM ∼ 4
SM
?
2. SA
? = ? = RI
3. SM =
4. T R =
5. 9? = 8?
Answer questions 6-9 about trapezoid ABCD.
6.
7.
8.
9.
10.
11.
Name two similar triangles. How do you know they are similar?
Write a true proportion.
Name two other triangles that might not be similar.
If AB = 10, AE = 7, and DC = 22, find AC. Be careful!
Writing How many angles need to be congruent to show that two triangles are similar? Why?
Writing How do congruent triangles and similar triangles differ? How are they the same?
Use the triangles below for questions 12-15.
AB = 20, DE = 15, and BC = k.
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12.
13.
14.
15.
Are the two triangles similar? How do you know?
Write an expression for FE in terms of k.
If FE = 12, what is k?
Fill in the blanks: If an acute angle of a _______ triangle is congruent to an acute angle in another ________triangle, then the two triangles are _______.
Use the diagram below to answer questions 16-20.
16. Draw the three separate triangles in the diagram.
17. Explain why 4GDE ∼ 4DFE ∼ 4GFD.
Complete the following proportionality statements.
18.
19.
20.
GF
DF
GF
GD
GE
DE
=
=
=
?
FE
?
GE
DE
?
Answers for Explore More Problems
To view the Explore More answers, open this PDF file and look for section 7.4.
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