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Bell Work • 1) A) Write a similarity statement for the similar figures
below:• B) What is the similarity ratio?• C) Solve for x and y
• 2) Simplify the following ratios:• A) B)
• 3) Write a statement of proportionality for the figures below:
ft
in
3
15
ft
yds
24
12
Agenda• 1) Bell Work
• 2) IP Check
• 3) Outcomes
• 6) 8.4 Notes
• 7) IP – Assessment #11
Outcomes
• I will be able to:
• 1) Use ratios and proportions to find missing side lengths of similar figures
• 2) Determine if triangles are similar
• 3) Use similarity shortcuts to determine if triangles are similar
Similar Triangle Activity1. Look at Triangle ABC and Triangle DEF: A (1, 1) and D (1, -1) B (4, 1) E (7, -1) C (4, 5) F (7, -9)
2. Look at Triangle JKL and Triangle MNO: J (1, 2) and M (-1, -1) K (5, 2) N (-3, -1) L (3, -2) O(-2, -3)
3. Look at Triangle PQR and Triangle STV: P (-2, 0) and S (-1, 1) Q (0, 4) T (0, 3) R (2, 0) V (1, 1)
Are these triangles similar? Yes/NoWhy?
Similarity Statement:
____________________
Statement of Proportionality?
__________________
What pieces did we need to determine similarity?
Results• 1. Look at Triangle
ABC and Triangle DEF where:
• A (1, 1) and D (1, -1)• B (4, 1) E ((7, -1)• C (4, 5) F (7, -9)• AB = ______ • DE = ______• m∠A = ___, m∠D=
____
• AC = _____ • DF = ______
• Are these triangles similar? Yes/No
• Why?
• Similarity Statement:
____________________
Statement of Proportionality?
__________________
• What pieces did we need to determine similarity?
Results• 2. Look at Triangle
JKL and Triangle MNO where:
• J (1, 2) and M (-1, -1)• K (5, 2) N (-3, -
1)• L (3, -2) O (-2, -
3)_____ and _____ MJ
_____ and _____ NK_____ and _____ OL
• Are these triangles similar? Yes/No
• Why?
• Similarity Statement:
____________________
Statement of Proportionality?
__________________
• What pieces did we need to determine similarity?
Results• 3. Look at Triangle
PQR and Triangle STV where:
• P (-2, 0) and S (-1, 1)
• Q (0, 4) and T (0, 3)• R (2, 0) and V (1, 1)
• PQ = ____ ST =_____• QR = ____ TV = ____• RP = _____ VS= ____
• Are these triangles similar? Yes/No
• Why?
• Similarity Statement:
____________________
Statement of Proportionality?
__________________
• What pieces did we need to determine similarity?
Why?• What did we find during this activity?
• Did we need to look at every single angle and every single side to determine if the triangles were similar?
• No, so there are triangle similarity shortcuts just as there were triangle congruence shortcuts.
Shortcuts
• Just as when looking at triangle congruence, there are shortcuts to help us determine if triangles are *similar.
Similar Triangles
If Two angles of one triangleare congruent to two anglesin another triangle, then the two triangles are similar.
Similarity Statement:
YXZKJL ~
Why do we only need two angles?
Examples
• 1) Explain why the triangles are similar and write a similarity statement
• Angle A is congruent• to Angle D• Angle C is congruent• in both triangles • because of vertical angles• So, by AA
B
AC
D
EDECABC ~
Examples
• 2) Are the following triangles similar?
• So, yes by the AA • similarity postulate
Sometimes you mayhave to solve for a piece that isn’t given to us in order to determine if they aresimilar.
x
x + 40 + 88 = 180x = 52
On Your Own• 3. Explain why the triangles are similar
and write a similarity statement• ΔRQP ~ ΔVUP by the AA postulate
because:
• Suppose RP = 15, RQ = 10 and UV = 7• Find VP and RV• What is the scale factor?
• Write a proportion to • solve:
PP
UQ
x710
15
10
7
1510
7 x
Similarity Theorems
If all the lengths of the corresponding sides of two triangles are proportional, then the triangles are similar.
If an angle of one triangle is congruent to an angle of a second triangle and the lengths of the sides including these angles are proportional, then the triangles are similar
1014
18
57
9
18
9
14
7
10
5*Check the side ratios:
612
816 95°
95°
*Check the side ratios and the angle between them
Examples
6
18
4
12
What should we do first?1. Label everything
What do we need to do next?
2. Look at what pieces of each triangle we actually have
We have two sides from eachso we can compare side ratios
12
4
18
6 So, we know two side ratios are the same. What
else do we know?Both triangles have Angle A, so it is congruent
So, these triangles are similar by SAS similarity shortcut
On Your OWN
Examples
60
48
40
52
65
x
1. Label what we know
2. Set up a proportion to find the missing piece.
What do we have to make sure wedo?
Compare pieces of the small trianglewith pieces of the big triangle
x
40
108
48 48x = 4320
x = 90m
Examples
1. Label what we know
2. What can we check?
6.32
6.38
6.27
6.32
22
26
***Be sure to create a proportion in which the corresponding parts line up
On Your Own