similarity day 1 sss, sas, aa
TRANSCRIPT
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#3.19 Geo. Drill 3/5/14•Given that the following two
pentagons are similar, find x.
x4
67
5
12 8
414
10
Turn in CW from
yesterday.
Pass out papers
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#3.19 Geo. Drill 3/5/14•Given that the following two
pentagons are similar, find x.
x4
67
5
12 8
414
10
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Geometry Drill
•Can you list the 5 ways to prove triangles congruent
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Objective•Students will use the similarity postulates to decide which triangles are similar and find unknowns.
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Congruence vs. Similarity
•Congruence implies that all angles and sides have equal measure whereas...
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Congruence vs. Similarity
•Similarity implies that only the angles are of equal measure and the sides are proportional
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AA Similarity Postulate
•Two triangles are similar if two pairs of corresponding angles are congruent
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AA Similarity Postulate
A
B
C D
E
F
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SAS Similarity Postulate
•Two triangles are similar if two pairs of corresponding sides are proportional and the included angles are congruent
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SAS Similarity Postulate
A
B
C D
E
F4 6
8 12
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SSS Similarity Postulate
•Two triangles are similar if all three pairs of corresponding sides are proportional.
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SSS Similarity Postulate
A
B
C D
E
F7
219 2713
39
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Example 1•∆APE ~ ∆DOG. If the perimeter of ∆APE is 12 and the perimeter of ∆DOG is 15 and OG=6, find the PE.
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A A by Reflexive Property of , and B C since they are both right angles.
Example 2
Explain why ∆ABE ~ ∆ACD, and then find CD.
Step 1 Prove triangles are similar.
Therefore ∆ABE ~ ∆ACD by AA ~.
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Step 2 Find CD.
Corr. sides are proportional. Seg. Add. Postulate.
Substitute x for CD, 5 for BE, 3 for CB, and 9 for BA.
Cross Products Prop. x(9) = 5(3 + 9)
Simplify. 9x = 60
Divide both sides by 9. 3
20
9
60x
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Explain why ∆RSV ~ ∆RTU and then find RT.
Step 1 Prove triangles are similar.
It is given that S T. R R by Reflexive Property of .
Therefore ∆RSV ~ ∆RTU by AA ~.
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Check It Out! Example 3 Continued
Step 2 Find RT.
Corr. sides are proportional.
Substitute RS for 10, 12 for TU, 8 for SV.
Cross Products Prop.
Simplify.
Divide both sides by 8.
RT(8) = 10(12)
8RT = 120
RT = 15
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Writing Proofs with Similar Triangles
Given: 3UT = 5RT and 3VT = 5ST
Prove: ∆UVT ~ ∆RST
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Statements Reasons
1. Given1. 3UT = 5RT
2. Divide both sides by 3RT.2.
3. Given.3. 3VT = 5ST
4. Divide both sides by3ST.4.
5. Vert. s Thm.5. RTS VTU
6. SAS ~ Steps 2, 4, 56. ∆UVT ~ ∆RST
Example 4 Continued
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Given: M is the midpoint of JK. N is the midpoint of KL, and P is the midpoint of JL.
Prove: ∆JKL ~ ∆NPM
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Statements Reasons
1. Given1. M is the mdpt. of JK, N is the mdpt. of KL,
and P is the mdpt. of JL.
2. ∆ Midsegs. Thm2.
3. Div. Prop. of =.3.
4. SSS ~4. ∆JKL ~ ∆NPM
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Example 5: Engineering Application
The photo shows a gable roof. AC || FG. ∆ABC ~ ∆FBG. Find BA to the nearest tenth of a foot.
From p. 473, BF 4.6 ft.
BA = BF + FA
6.3 + 17
23.3 ft
Therefore, BA = 23.3 ft.
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Check It Out! Example 5
What if…? If AB = 4x, AC = 5x, and BF = 4, find FG.
Corr. sides are proportional.
Substitute given quantities.
Cross Prod. Prop.
Simplify.
4x(FG) = 4(5x)
FG = 5
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Conclusion•Similarity•AA•SAS•SSS
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Classwork/Homework
•Page 474-475 #’s 1-10, 17,18, 23, 24, 44-46