Download U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb

U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
Unit 5 Lesson 7: Proving Triangles Similar
This lesson gives us an understanding of the different and most efficient ways that we can prove triangles to be similar to each other. These 2 slides explain "WHY" AA is sufficient for triangle similarity!
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.
When studying congruent triangles in G.CO.8 we established some minimum requirements that would guarantee congruence through a single or sequence of isometric transformations. We found that SSS, SAS, ASA, AAS and HL (and some special cases of ASS) to be enough information to always establish congruence between two triangles. In a likewise manner, we want to find the minimum requirements in two triangles to establish similarity. The way we are going to do this is to use a single or sequence of similarity transformations that would map one triangle onto the other.
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
Teacher to explain
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NOTE: This is the most common way to prove triangles similar as we will see in the near future
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...in short
Angle‑Angle (AA) Similarity Postulate
If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar.
If <A ≅ <D and <B ≅ <E, then ΔABC ~ ΔDEF.
C
A
E
D
F
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
C
The triangles shown are similar. 1) Which angles are ≅?
B
A
D
E
2) Write a statement of similarity.
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3) Write a statement of proportionality.
The triangles shown are similar.
1) Which angles are ≅?
A
B
D
C
E
2) Write a statement of similarity.
3) Write a statement of proportionality.
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
!!
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DO
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Use the diagram to complete the following:
1) ΔPQR ~ _____
Q
y
P
20
2) PQ = QR = RP
?
?
?
x
R
3) 20 = ?
?
12
4) ? = 18
20 ?
5) x = _____
12
L
M
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18
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6) y = ____
Use the properties of similarity transformations to establish the SSS criterion for two triangles to be similar.
∆ABC ∼ ∆DEF because ∆ABC was mapped onto ∆DEF using only similarity transformations. Thus SSS is a similarity criterion.
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
Side-Side-Side (SSS) Similarity Theorem:
If the corresponding lengths of the
sides of two triangles are proportional,
then the triangles are similar.
A
P
If AB = BC = CA
PQ
QR
RP
then ΔABC ~ ΔPQR
C
B
Q
R
Now let us see if knowing two corresponding proportional sides and the included corresponding congruent angle (SAS) is enough for establishing similarity. Just as it was important in congruence the angle must be the included angle – meaning that it is the one between the two sides.
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
A
Side-Angle-Side (SAS) Similarity Theorem:
If an angle of one triangle is congruent
to an angle of a second triangle, and
the lengths of the sides including these
angles is proportional, then the
triangles are similar.
If <A ≅ <P and AB = AC
PQ PR
P
C
B
Q
R
then ΔABC ~ ΔPQR
Prove: ΔABC ~ ΔEDC
statements
1)
reasons
1) Given
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
statements
1)
1) Given
statements
1)
reasons
reasons
1) Given
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
Sometimes we prove similarity to establish new relationships about the triangle. !
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loo
Once similarity is established we know that there are three corresponding congruent angles and that the sides are proportional. This is a bit like congruence where once we had proven the triangles to be congruent we could use CPCTC (Corresponding Parts of Congruent Triangles are Congruent) to gather new relationships about the triangle.
statements
1)
reasons
1) Given
Look at what we have to prove now!!
statements
1)
reasons
1) Given
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U5 L7 Proving Triangles Similar For Wed Feb 15th and Thurs Feb 16th.notebook
AB DE
1
2
4
statements
1)
reasons
3
1) Given
Homework U5 L7 Day #1 for Wed 2/15
Homework U5 L7 Day #2 for Thursday 2/16
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