Download Angle-Angle and Side-Side-Side Similarity Theorems

Angle-Angle and
Side-Side-Side Similarity
Theorems
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Printed: August 11, 2012
AUTHORS
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C ONCEPT
Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems
1
Angle-Angle and
Side-Side-Side Similarity Theorems
Lesson Plan
Launch (10 min):
• Students will study the ASS and AAA cases for why they don’t work
• Teacher summarizes
• Transition into similar triangles off of AAA example
Presentation (10 min):
• Go over congruency vs. similarity definitions using a t-chart to contrast
• Highlight each corresponding side pair of angles or sides and then highlight that same pair in the ratio below
it
Practice (30 min): AA and SSS
• Go over class examples and non-examples for both AA and SSS similarity.
• Emphasize the stating of congruent angles in AA and the correct setup of ratios and their comparison in SSS.
Conclusion (10 min):
• Students will create their own examples of SSS congruence and SSS similarity. They will then use a word
bank to help them explain the difference between them.
Exit Ticket (5 min):
Homework
Materials:
• Rulers
• Protractors
Launch
Final Congruency Investigations
Part 1
1
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TABLE 1.1:
Marking S/A?
Congruency
Name the 3-letter congruencies group given in both triangles: ____ ____ ____
∗ Are
the two triangles congruent? Yes/No
∗ Does
ASS/SSA make only 1 kind of triangle? Yes/No
∗ Therefore,
if we have ASS/SSAare we sure that we have two congruent triangles? Yes/No
Part 2
Draw the following 2 equiangular triangles (each angle will be 60 degrees). Label their sides and angles.
TABLE 1.2:
Triangle 1
Side lengths 2 cm, 2cm, 2cm
Triangle 2
Side lengths 6 cm, 6 cm, 6 cm
∗ Are
the triangles congruent? Yes/No
∗ Can
we make only 1 kind of triangle with AAA given information? Yes/No
∗ Therefore,
is AAA enough given info to know for sure that 2 triangles are congruent? Yes/No
Circle the acceptable groups of evidence to say two triangles are congruent:
SSS
SAS
ASS
ASA
AAS
AAA
Presentation
Similarity (∼) Two triangles are similar if their 3 corresponding angle pairs are congruent and their 3 corresponding
side pairs are proportional (same scale factor)
TABLE 1.3:
Congruent
∼
=
Identical shape & size
Angles: Congruent
Sides: Congruent
2
Similar
∼
Identical shape;
different sizes possible
Angles: Congruent
Sides: Proportional (scale factor)
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Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems
TABLE 1.3: (continued)
Congruent
Similar
∆STU ∼
= ∆XWV
∆STU ∼ ∆XWV
Scale Factor: 3
√
6 √3
∆1
XW
=
=3
∆2
ST
2 3
VX
US
=
6
2
=3
WV
TU
=
3
1
=3
Similarity Evidence Groups
AA Similarity → If two angles in one triangle are _________ to two angles in another, then the triangles are
________. (Note: if two angle pairs are congruent, then the third angle pair must also be _______).
SSS Similarity → If all 3 corresponding sides between two triangles are _________, then the triangles are ___________.
Practice
Directions: Are the triangles similar? Say yes or no and provide evidence. If yes, make a similarity statement like
the ‘Examples’ above.
1)
3
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Yes/No
∆
∼∆
by _____
2)
Yes/No
∆
∼∆
by _____
3)
Yes/No
∆
4)
4
∼∆
by _____
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Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems
∆1
∆2
Yes/No
∆
∼∆
by _____ (Scale factor _____)
5)
∆1
∆2
Yes/No
∆
∼∆
by _____ (Scale factor _____)
6)
∆1
∆2
5
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Yes/No
∆
∼∆
by _____ (Scale factor _____)
7)
Can you have two triangles that are both congruent and similar? Use the triangles at the left and your ratio tests for
similarity to answer the question.
Conclusion
1. Sketch and label an example of SSS Congruence.
2. Sketch and label an example of SSS Similarity.
What is the difference between the two? Explain using at least 2 complete sentences and 2 of the words below:
Triangles
Sides
Angles
Congruent
Similar
Scale Factor
Size
Homework
1. Two triangles have the following side lengths: 3 cm, 5 cm, 7 cm and 6 cm, 10 cm, and 14 cm. Are the two
triangles similar? Provide evidence.
Similar? Say yes or no and provide evidence. If yes, make a similarity statement.
2.
6
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Concept 1. Angle-Angle and Side-Side-Side Similarity Theorems
3.
4.
5.
7
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Exit Ticket
8