1. Unit: Triangles

2. 3-4 Parallel lines and Triangle Sum Theorem
Objective: To classify triangle and find the measure of their angles To use exterior angle Theorem

3. Classifying Triangle
Classify by angles: Classify by sides:

4. Theorems
Triangle-Angle Sum Theorem: Sum of the angles is 180. Exterior Angle Theorem: sum of the remote interior angles equals the exterior angle C + B = BAD
Remote interior angles
Exterior angle

5. Example 1 Using the Exterior Angle Theorem
63+56 = 119
X = 119

6. Example 2: Exterior angle & Sum of the angle of a triangle
90-55 = 35
86-55 = 31
180-(86+35) = 59

7. TRY Find the measure of each angle. 62-25 = 37 180-(56+62) =62
180 – 62 = 118 OR
= 118

8. Example 3 Classify the triangles. By its sides 18cm, 20 cm, 18cm
isosceles
b) By its angles 91,20 ,69
obtuse

9. Try Classify the triangle. The measure of each angle is 60.
Equilateral and equiangular

10. Closure What is the sum of the interior angles of a triangle? 180
2) What is the relationship of the exterior-angle and the two remote interior angles?
Sum of the remote interior angles = exterior angle.

11. 3-5 Polygon Angle Sum
Objective: To classify polygons To find the sums of the measures of the interior and exterior angles of a polygon

12. Vocabulary Polygon Concave Convex Equilateral polygon
closed figure with the at least three segments.
Concave Convex
Equilateral polygon
All sides congruent
Equiangular polygon
All angles congruent
Regular polygon
Equilateral and equiangular polygon
convex
convex
concave
concave
convex

14. Polygon Names Name Number of sides Sum of the interior angles
An interior angles
An exterior angles of a regular polygon
Triangle 3 60
120
Quadrilateral 4
360 90
Pentagon 5
540
108 72
Hexagon 6
720
Heptagon 7
900
900/7
51 3 /7
Octagon 8
1080
135 45
Nonagon 9
1260
140 40
Decagon 10
1440
144 36
Unagon 11
1620
1620/11
32 /11
Dodecagon 12
1800
150 30
N-gon n
(n-2)180
360/n

15. Polygons
Polygon Angle Sum Theorem 180(n-2) Polygon Exterior Angle Theorem: Sum of all exterior angles is 360 degrees.

16. Example 1 Name the polygon by its sides Concave or convex.
Pentagon
Name the polygon by its sides
Concave or convex.
Name the polygon by its vertices.
Find the measure of the missing angle
Convex
QRSTU
(5-2)180 = 540
= 411
540 – 411 = 129

17. Example 2
Find the measure of an interior and an exterior angle of the regular polygon..
(7-2)180/7 = 128 4/7
360/7 = /7

18. Determine the number of Sides
If the sum of the interior angles of a regular polygon is 1440 degrees.
1440 = 180(n-2)
8 = n-2
10 = n it is a decagon
Find the measure of an exterior angle
360/10 = 36 degrees

19. Closure
What is the formula to find the sum of the interior angles of a polygon?
(n-2)180
What is the name of the polygon with 6 sides?
hexagon
How do you find the measure of an exterior angle?
Divide the 360 by the number of sides.

20. 4-5 Isosceles and Equilateral Triangles
Objective: To use and apply properties of isosceles and equilateral triangles

21. Isosceles Triangle Key Concepts
Isosceles Triangle Theorem
Converse of the Isosceles Triangle
Theorem

22. Isosceles Triangle Key Concepts
If a segment, ray or line bisects the vertex angle, then it is the perpendicular bisector of the base.

23. Equilateral Triangle Key Concepts
If a triangle is equilateral,
then it is equiangular.
If the triangle is equiangular,
then it is equilateral.

24. What did you learn today?
What is still confusing?

25. 5-1 Midsegments
Objective: To use properties of midsegments to solve problems

26. Key Concept
Midsegments – DE = ½AB and DE || AB

27. Try 1
Find the perimeter of ∆ABC = 42

28. Try 2: If mADE = 57, what is the mABC? 57°
b) If DE = 2x and BC = 3x +8, what is length of DE?
4x = 3x+8
x = 8
DE = 2(8) = 16

29. What have you learned today? What is still confusing?

30. 7-1 Ratios and Proportions
Objective: To write ratios and solve proportions.

31. VOCBULARY RATIO- COMPARISON OF TWO QUANTITIES.
PROPORTION- TWO RATIOS ARE EQUAL.
EXTENDED PROPORTION – THREE OR MORE EQUILVANT RATIOS.

32. PROPERTIES OF RATIOS
a c is equivalent to: 1) ad = bc b d 2) b d 3) a b a c c d 4) a + b c + d b c

33. Example 1 5 20 x 3 b) 18 6 n + 6 n 15 = 20x ¾ = x 18n = 6n +36 x b)
n n
15 = 20x
¾ = x
18n = 6n +36
12n = 36
n = 3

34. Example 2 /8 x
X = 16 (7/8)
X = 14 ft
The picture above has scale 1in = 16ft to the actual water fall
If the width of the picture is 7/8 inches, what is the size of the actual width of the part of the waterfall shown.

35. 7-2 Similar Polygons
Objective: to identify and apply similar polygons

36. Vocabulary
Similiar polygons- (1) corresponding angles are congruent and (2) corresponding sides are proportional. ( ~)
Similarity ratio – ratio of lengths of corresponding sides

37. Example 1 Find the value of x, y, and the measure of angle P.
4/6 = 7/Y X/9 = 4/6
4Y = X = 36
Y = X = 6

38. Example 2
Find PT and PR 4 = X 11 X+12 11X = 4X X = 48 X = 6 PT = 6 PR = 18

39. Example 3
Hakan is standing next to a building whose shadow is 15 feet long. If Hakan is 6 feet tall and is casting a shadow 2.5 feet long, how high is the building? X = X = 80 X =

40. TRY
A vertical flagpole casts a shadow 12 feet long at the same time that a nearby vertical post 8 feet casts a shadow 3 feet long.  Find the height of the flagpole.  Explain your answer. 

41. 5-2 Bisectors in Triangles
objective: To use properties of perpendicular bisectors and angle bisectors

42. Key Concept
Perpendicular bisectors – forms right angles at the base(side) and bisects the base(side). Angle Bisectors– bisects the angle and equidistant to the side.

43. Try 1
WY is the  bisector of XZ Isosceles triangle

44. Try 2
6y = 8y -7 7 = 2y y = 7/2 21 Right Triangle

45. What have you learned today? What is still confusing?

46. 5-3 Concurrent Lines, Medians, and Altitudes
Objective:
To identify properties of perpendicular bisectors and angle bisectors
To Identify properties of medians and altitudes

47. Key Concept
Perpendicular Bisectors Altitudes circumscribe Medians Angle Bisectors inscribe

48. Key Concepts
Medians – AD = AG + GD AG = 2GD
E F D

49. Try
Give the coordinates of the point of concurrency of the incenter and circumcenter.
Angle bisectors ( 2.5,-1)
Perpendicular bisectors
(4,0)

50. Try Give the coordinates of the center of the circle.
(0,0) perpendicular bisectors.

51. Determine if AB is an altitude, angle bisector, median, perpendicular bisector or none of these?

52. What have you learned today? What is still confusing?

53. 7-5 Proportions in Triangles
Obj: To use the Side-Splitter Theorem and Triangle-Angle Bisector Theorem.

54. Side-Splitter Theorem
If a line is parallel to a side of a triangle and intersect the other two sides, then this line divides those two sides proportionally.
Side-Splitter Theorem

55. Triangle-Bisector Theorem
if a ray or segment bisects an angle of a triangle then divides the segments proportionally. Triangle-Angle Bisector Theorem

56. Example 1 Find the value of x. 24 40 x 30 24 = x 40 30 720 = 40x
720 = 40x
18 = x

57. Example 2 Find x and y. 6 = 5 x = 12.5 X 12.5 9 y X = 15 y = 7.5 6 5 x y
6 = x = 12.5
X y
X = 15 y = 7.5

58. What have you learned today? What is still confusing?

59. 5-5 Inequalities in Triangles
Objective:
To use inequalities involving angles of triangles
To use inequalities involving sides of triangles

60. Key Concepts
Triangle inequality – the sum of two sides is greater than the third side.

61. Try Order angles from least to greatest. B, T, A
Order the sides from lest to greatest.
BO, BL, LO

62. Try
Can the triangles have the given lengths? Explain. yes > 8 yes > 9 yes < 4.9 no

63. Try Describe possible lengths of a triangles. 4in. and 7 in
7 –
< third side length < 11
3 < x < 11

64. What have you learned today? What is still confusing?

65. Simplifying Radicals √ radical Radicand – number inside the radical
You can click on other videos for more explainations. √

66. Examples √6 ∙ √8 √2∙2∙2∙2∙3 4√3 2) √90 √2∙3∙3∙5 3√10
√6 ∙ √8
√2∙2∙2∙2∙3
4√3
2) √90
√2∙3∙3∙5
3√10
3) √243 √3 √3∙3∙3∙3∙3 9 √3 9

67. Division – multiply numerator and denominator by the radical in the denominator
4) √25 √3
5 ∙√3
√3 ∙√3 3
= √14 √
6) √5 ∙ √35 √14 √5∙5 ∙7 √2∙7 5√7 √2∙7 √2∙7 √2∙7 35 √2 = 5 √2 14 2

68. What have you learned today? What is still confusing?

69. Chapter 8-1 Pythagorean Theorem and It’s Converse
Objective: to use the Pythagorean Theorem and it’s converse. c2 = a2 + b2

70. Pythagorean Triplet
Whole numbers that satisfy c2 = a2 + b2. Example: 3, 4, 5 Can you find another set?

71. Ex 1 Find the value of x. Leave in simplest radical form.
Answer: 2 √11 x 12 10

72. Ex 2: Baseball
A baseball diamond is a square with 90 ft sides. Home plate and second base are at opposite vertices of the square. About far is home plate from second base? About 127 ft

73. Pythagorean Theorem Acute c2 < a2 + b2 Right c2 = a2 + b2
C b A
Acute c2 < a2 + b2
Right c2 = a2 + b2
Obtuse c2 > a2 + b2 B
a c
C b A B
a c
C b A

74. Ex 3:Classify the triangle as acute, right or obtuse.
15, 20, 25
right
b) 10, 15, 20
Obtuse

75. What have you learned today? What is still confusing?

76. Ch 8-2 Special Right Triangles
Objective: To use the properties of 45⁰ – 45⁰ – 90⁰ and 30⁰ – 60⁰ - 90⁰ triangles. 45⁰ – 45⁰ – 90⁰ 30⁰ – 60⁰ - 90⁰ x - x - x√2 x - x√3 - 2x

77. Special Right Triangles
45⁰ – 45⁰ – 90⁰ ⁰ – 60⁰ - 90⁰

78. Example 1
Find the length of the hypotenuse of a 45⁰ – 45⁰ – 90⁰ triangle with legs of length 5√6 . 45⁰ – 45⁰ – 90⁰ x - x - x√2 X = 5√6 x√2 = 5√6√2 substitute into the formula = 10 √3

79. Example 2
Find the length of a leg of a 45⁰ – 45⁰ – 90⁰ triangle with hypotenuse of length ⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 22 solve for x X = 22 = 22√2 = 11√2 √2 2

80. Example 3:
The distance from one corner to the opposite corner of a square field is 96ft. To the nearest foot, how long is each side of the field? 45⁰ – 45⁰ – 90⁰ x - x - x√2 x√2 = 96 solve for x X = 96 = 96√2 = 48√2 √2 2

81. Example 4
The longer leg of a 30⁰ – 60⁰ - 90⁰ triangle has length of 18. Find the lengths of the shorter led and the hypotenuse. 30⁰ – 60⁰ - 90⁰ x - x√3 - 2x x√3 = 18 solve for x X = 18 = 18√3 = 6√3 – short leg √3 3 12√3 - hypotenuse

82. Example 5 Solve for missing parts of each triangle: x = 10 y = 5√3 x y

83. What have you learned today? What is still confusing?

84. 7-4 Similarities in Right Triangles
Objective: To find and use relationships in similar right triangles

85. Geometric mean with similar right triangles

86. Example 1 Find the Geometric Mean of 3 and 15. √3∙15 3 √ 5
√3∙48 12

87. Example2 Find x, y, and z. X = 6 9 x 36 = 9x 4 = x 9 = z z 9+x
y = x
9+x y
Y ² = 4(13)
Y = 2√13