This document provides information on ratios, proportions, and similarity between triangles: 1) It defines ratios, proportions, and the properties of proportions including the cross product property and reciprocal property. 2) It explains the conditions for triangles to be similar - if the corresponding angles are congruent (ASA, SAA) or if the corresponding sides are proportional (SSS). 3) It includes examples of using proportions to find unknown values and determining if two triangles are similar based on given information about their sides and angles.

1. RatioandProportion

3. LearningTargets
Expressratioinsimplestform.
Solvefortheunknownterminagivenproportion.
•Showappreciationoftheapplicationofsimilarity
insolvingreal-lifeproblems.

4. Simplifytheratios:
1.
15
45
2.
15𝑐𝑚
3𝑚
3.
34
74

5. Ratiowrittenin fourways
a:b
atob
a/b
𝑎
𝑏

6. Proportion
Anequationthatequatestworatios.
Eachnumberinaproportioniscalleda
term.Intheproportiona:b=c:d,theterms
aanddarecalledextremeswhilethe
termsbandcarecalledmeans.

7. PropertiesofProportion
1. CrossProductProperty:Theproductpfthe
meansequalstheproductoftheextremes.
2. ReciprocalProperty:Iftworatiosareequal,
thentheirreciprocalsarealsoequal.

8. Findtheunknownintheproportion
8
𝑏
=
𝑏
12
(8)(12) (b)(b)
=
60 = 𝑏2
60 = 𝑏2

9. 8:7=40:x
(8) (x) = (7)(40)
8x = 280

10. 7x-30 : 9x-30 = 5 : 7
(7x-30) (7) = (9x-30)(5)
49x – 210 = 45x - 150
49x – 45x = 210 - 150
4x = 60
x = 15

11. (6)(7) = 2 (y – 3)
42 = 2y - 6
42 + 6 = 2y
48 = 2y
y
=
24

12. (x)(16) = (8)(5)
16x = 40
x = 2.5

13. 4 (x + 3) = 7 (x – 2)
4x + 12 = 7x -14
12 + 14 = 7x -4x
26 3x
=

14. 4 (x + 3) = 7 (x – 2)
4x + 12 = 7x -14
12 + 14 = 7x -4x
26 3x
=

15. Workbook page 79

16. Similarity Between
Triangles

18. LearningTargets
- Illustrate similarities of figure.
- Prove the conditions for similarity of
triangles.
- Determine if two triangles are similar

19. ASA

20. SSS

21. SAA

22. SimilarTriangles
Whenthereisacorrespondingbetween
twotrianglessuchthattheir
correspondinganglesarecongruentand
thelengthsofthecorresponding sides
areproportional.

25. Inthefigure△ABC ∼ △DEF , complete the
following statements given that AB = 6, BC =
8, AC = 10 and DE = 3.
Find the following:
𝐷𝐸
𝐴𝐵
=?
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟△DEF
𝑃𝑒𝑟𝑖𝑚𝑒𝑡𝑒𝑟△ABC
=?

26. ScaleFactor
Theratioofthelengthsoftwocorrespondingsides.

27. Postulate:AAASimilarity
Postulate
Two triangles are similar if and only if
their corresponding angles are
congruent.
Consider △ABC and △EFG . If ∠A ≅
∠E, ∠B ≅ ∠F, and ∠C ≅ ∠G, then we
conclude that △ABC ∼ △EFG .

29. Theorem:AASimilarityTheorem
Iftwoanglesofonetrianglearecongruentto
twoanglesofanothertriangle,thenthe
trianglesaresimilar.

31. Theorem:TheSASSimilarityTheorem
Ifintwotrianglesthetwopairsof
correspondingsidesareproportional
andtheincludedanglesare
congruent,thenthetwotrianglesare
similar.

32. 𝑃𝑄
𝑋𝑌
𝑄𝑅
𝑌𝑍
=
𝑃𝑅
𝑋𝑍
=

33. Determineifthetwogiventrianglesare
similar

34. Determineifthetwogiventrianglesare
similar

35. Theorem:TheSSSSimilarityTheorem
Ifthecorrespondingsidesoftwo
trianglesareproportional,then
thetrianglesaresimilar.

36. 𝐴𝐶
𝐷𝐹
=
10
15
=
2
3
𝐶𝐵
𝐹𝐸
=
8
12
=
2
3
𝐴𝐵
𝐷𝐸
=
12
18
=
2
3

37. 𝐴𝐶
𝐷𝐸
=
8
6
=
4
3
𝐶𝐵
𝐹𝐸
=
16
12
=
4
3
𝐴𝐵
𝐷𝐸
=
12
9
=
4
3

38. ACTIVITY

39. ACTIVITY

40. ACTIVITY

41. Similarities in Right
Triangles

42. LearningTargets

45. Other Theorems on Right
Triangles

46. LearningTargets
StateandprovethePythagorean
Theorem;
Stateandprovethetheoremsonspecial
righttriangle;and
Solveproblemsthatinvolvetriangle
similarityandrighttriangles.

47. PythagoreanTheorem
Inarighttriangle, thesquareofthehypotenuse isequalto thesum ofthesquares of the
legs.