1. Welcome to the Math S.A.T. Enjoyment Hours

2. Hosted by the B B & S Brothers Bianco, Bianco & Skeels

3. Quick Drillsky

4. #

5. #2 180 ÷ 3

6. #

7. #4 (12)2

8. #5 (2)5

9. #6 (10)8

10. #7 √ 169

11. #8 √ (475)2

12. #9 (9)9 (3)18

13. #

14. LET’S √ EM!

15. #

16. # 90

17. #2 180 ÷ 3

18. #2 180 ÷ 3 60

19. #

20. # 49

21. #4 (12)2

22. #4 (12)2
144

23. #5 (2)5

24. #5 (2)5 32

25. #6 (10)8

26. #6 (10)8
100,000,000

27. #7 √ 169

28. #7 √ 169 13

29. #8 √ (475)2

30. #8 √ (475)2
475

31. #9 (9)9 (3)18

32. #9 (9)9 (3)18 1

33. #

34. #
180

35. You can have PSAT/SAT Fun everyday! Go to www.collegeboard.com

36. Strategy !
If the sum of 4 consecutive integers is ‘f’, then, in terms of ‘f’, what is the least of these integers?
A) f/4 B) (f - 2)/4 C) (f - 3)/4 D) (f - 4)/4 E) (f - 6)/4

37. Strategy - Substitute!
If the sum of 4 consecutive integers is ‘f’, then, in terms of ‘f’, what is the least of these integers?
A) f/4 B) (f - 2)/4 C) (f - 3)/4 D) (f - 4)/4 E) (f - 6)/4

38. Strategy - sdrawkcaB kroW
Work backwards!!!! Fill in the answer choices for complex algebra problems.
Example: If (a/2)3 = a2, a≠0, then a =
A) 2 B) C) 6 D) E) 10
*From last lesson - ran out of time!

39. Helpful Hint:
Remember the answer choices are arranged from least to greatest so it may help start in the middle and proceed in the right direction.

40. Objectives: To review Geometry concepts on SAT.
To introduce Student Produced Response problems.(SPR)
To introduce 1 more strategy.

41. GEOMETRY & MATH
WE ALL KNOW FIGURES INVOLVED IN GEOMETRY

42. GEOMETRY & MATH
WE ALL KNOW FIGURES INVOLVED IN GEOMETRY

43. GEOMETRY & MATH
WE ALL KNOW FIGURES INVOLVED IN GEOMETRY

44. GEOMETRY & MATH
BUT WITH A FEW DEFINITIONS WE CAN TACKLE MANY PROBLEMS WHICH OTHERWISE WOULD BE IMPOSSIBLE

45. ESSENTIALS OF GEOMETRY
A RIGHT ANGLE:

46. ESSENTIALS OF GEOMETRY
A RIGHT ANGLE: An angles with a measure of 90°

47. ESSENTIALS OF GEOMETRY
AN ACUTE ANGLE:

48. ESSENTIALS OF GEOMETRY
AN ACUTE ANGLE: An angle which measurement is less than 90°

49. ESSENTIALS OF GEOMETRY
AN OBTUSE ANGLE:

50. ESSENTIALS OF GEOMETRY
AN OBTUSE ANGLE: An angle which measurement is more than 90°

51. ESSENTIALS OF GEOMETRY
PERPENDICULAR LINES:

52. ESSENTIALS OF GEOMETRY
PERPENDICULAR LINES: Two lines that intersect at right angles ( note written as )

53. ESSENTIALS OF GEOMETRY
VERTICAL ANGLES: 2 1

54. ESSENTIALS OF GEOMETRY
VERTICAL ANGLES: Two intersecting lines form 2 pair of vertical angles.

55. ESSENTIALS OF GEOMETRY
VERTICAL ANGLES: 1 2
1 and 2 are vertical

56. ESSENTIALS OF GEOMETRY
VERTICAL ANGLES: ALWAYS HAVE THE SAME MEASURE!

57. ESSENTIALS OF GEOMETRY
SUPPLEMENTARY ANGLES : 2 1

58. ESSENTIALS OF GEOMETRY
SUPPLEMENTARY ANGLES : Two angles whose measures have a sum of 180°

59. ESSENTIALS OF GEOMETRY
COMPLEMENTARY ANGLES : 1 2

60. ESSENTIALS OF GEOMETRY
COMPLEMENTARY ANGLES : Two angles whose measures have a sum of 90°

61. ESSENTIALS OF GEOMETRY
SUM OF THE ANGLES IN A TRIANGLE: 1 2 3

62. ESSENTIALS OF GEOMETRY
SUM OF THE ANGLES IN A TRIANGLE: The sum of the three angles in a triangle is 180°

63. ESSENTIALS OF GEOMETRY
SUM OF THE ANGLES IN A TRIANGLE: 1
m+ m + m = 180 2 3

64. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

65. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c
a2 + b2 = c2 a b

66. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

67. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

68. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

69. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

70. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

71. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

72. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

73. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

74. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

75. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

76. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c a b

77. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM: c a b

78. ESSENTIALS OF GEOMETRY
PYTHAGOREAN THEOREM:
NOTE C IS ALWAYS OPPOSITE THE RIGHT ANGLE c A B

79. GEOMETRY PRACTICE
FIND THE VALUE OF X X 3 4

80. GEOMETRY PRACTICE
FIND THE VALUE OF X
a2 + b2 = c2 X 3 4

81. GEOMETRY PRACTICE
FIND THE VALUE OF X
a2 + b2 = c2 X 3
= X2 4

82. X 3 4 GEOMETRY PRACTICE a2 + b2 = c2 32 + 42 = X2 9 + 16 = X2
FIND THE VALUE OF X
a2 + b2 = c2 X
= X2 3
= X2 4

83. X 3 4 GEOMETRY PRACTICE a2 + b2 = c2 32 + 42 = X2 9 + 16 = X2 25= X2
FIND THE VALUE OF X
a2 + b2 = c2 X
= X2 3
= X2 4
25= X2

84. ±5 = X X 3 4 GEOMETRY PRACTICE FIND THE VALUE OF X A2 + B2 = C2

85. X = 5 X 3 4 GEOMETRY PRACTICE FIND THE VALUE OF X A2 + B2 = C2

86. Geometry Tips
Often figures are not drawn to scale. Redraw the diagrams more accurately.

87. Geometry Tips
Sometime it is helpful to add extra segments, lines, etc. to a drawing.

88. 1,000,000 words (inflation) Geometry Tips
If there is no drawing, make your own. A picture is worth what?
1,000,000 words (inflation)

89. 133° x° GEOMETRY PRACTICE Find the value of x: A 37 B 47 C 57
D 90 E 133
133° x°

90. 133° x° 133° GEOMETRY PRACTICE
First you must realize that angle 133° and the angle x° are supplementary angles

91. GEOMETRY PRACTICE
133° x°
133°
Then let: x° + 133° = 180°

92. 133° x° 133° GEOMETRY PRACTICE Then let: x° + 133° = 180°
Subtract: ° °

93. 133° x° 133° GEOMETRY PRACTICE Then let: x° + 133° = 180°
Subtract: ° °
Finally : x° = 47°

94. 133° x° GEOMETRY PRACTICE Find the value of x: A 37 B 47 C 57
D 90 E 133
133° x°

95. GEOMETRYPRACTICE
Find the value of x:
A 23 B 33 C 43 D57 E 90 x°
57°

96. 90° GEOMETRY PRACTICE Find the value of x: x° 57°
A 23 B 33 C 43 D57 E 90 x°
90°
57°

97. GEOMETRY PRACTICE x°
57°
x° + 57°+ 90° = 180°

98. GEOMETRY PRACTICE x°
57°
x° + 147° = 180°

99. GEOMETRY PRACTICE x°
57°
x° + 147° = 180°
-147° °

100. GEOMETRY PRACTICE x°
57°
x° + 147° = 180°
-147° °
x° = 33°

101. GEOMETRYPRACTICE
Find the value of x:
A 23 B 33 C 43 D57 E 90 x°
57°

102. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 x 8 12

103. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 y x 8 12

104. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 y x 8 12
82 + y2 = 172

105. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 y x 8 12
y = 15

106. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 x 15 8 12

107. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 x 15 8 12
x = 152

108. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 x 15 8 12
x = 152

109. GEOMETRY PRACTICE
FIND THE VALUE OF X 17 x 15 8 12
x = 9

110. GEOMETRY PRACTICE
The complement of an angle is 44more than the angle. What is the sum of the angle’s complement and its supplement?

111. x + 44 x

112. x + x + 44 = 90
x + 44 x

113. x + x + 44 = 90
2x + 44 = 90
x + 44 x

114. x + x + 44 = 90
2x + 44 = 90
2x = 46
x + 44 x

115. x + x + 44 = 90
2x + 44 = 90
2x = 46
x = 23
x + 44 x

116. GEOMETRY PRACTICE
The complement of an angle is 44more than the angle. What is the sum of the angle’s complement and its supplement?

117. Solution Angle is 23 Complement is 90 - 23 = 67
Supplement is = 157
Sum of comp & supp is 224

118. GEOMETRY Coordinate Geometry Lines and angles Triangles and Polygons
Perimeter
Area
Volume

119. Coordinate Geometry
Distance formula: d = √(x2 - x1)2 + (y2 - y1)2

120. Coordinate Geometry Distance formula: d = √(x2 - x1)2 + (y2 - y1)2
Slope: ∆y = (y2 - y1) ∆x (x2 - x1)

121. Lines and Angles
Adjacent angles 3 2 4 1

122. Lines and Angles
Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4 3 2 4 1

123. Lines and Angles Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4 Vertical angles

124. Lines and Angles Adjacent angles - 2,3 ; 3,4 1,2 ; 1,4
Vertical angles 1,3 ; 2,4 3 2 4 1

125. Parallel Lines: m || n 1 5 m 2 6 3 7 n 4 8 t

126. Triangles
Interior angles always have a sum of

127. Triangles Interior angles always have a sum of 180°.
Exterior angles always have a sum of

128. Triangles Interior angles always have a sum of 180°.
Exterior angles always have a sum of 360°. (1 at each vertex)
Each exterior angle is equal to the sum of the 2

129. Triangles Interior angles always have a sum of 180°.
Exterior angles always have a sum of 360°. (1 at each vertex)
Each exterior angle is equal to the sum of the 2 remote interior angles.
Similar triangles have corresponding sides which are proportional. (CSSTP)

130. Triangles Area of a ∆ = 1/2 base times height
∆ Inequality Thm - The sum of any two lengths must be greater than the third length.
Isosceles ∆- 2 or more congruent sides. (Angles opposite those sides are also congruent.)
Equilateral ∆ - all sides and angles are congruent.

131. Right Triangles a b c
Pythagoras said “In a right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
or a2 + b2 = c2

132. Rt. ∆s - Perfect Triples a b c
3, 4, 5;
5,12,13;
8, 15, 17
7, 24, 25

133. Rt. ∆s - Perfect Triples 3, 4, 5; 5,12,13; 8, 15, 17 7, 24, 25a b c
3, 4, 5;
5,12,13;
8, 15, 17
7, 24, 25
All multiples of these are also perfect triples.

134. Special Right Triangles
1, √3, 2
1, 1, √2 ∆ ∆ 2x
x√2
60° x x
30° x
x√3

135. Other Polygons Define and give area for each. Parallelogram Rectangle
Square
The sum of the interior angles for any convex polygon is

136. CIRCLES Circumference C = 2πr Area A = πr2
Arc lengths and sectors, multiply by portion of circumference or area used.

137. SOLIDS Surface area and Volume Use formula sheet.
Know these before the test.

138. Strategy:BoD
On Geometry problems be careful of figures that are “not drawn to scale”, redraw as acurate a figure as you can. Feel free to extend lines, rays, etc., or draw extra segments as needed.
Example: Find the value of x.

139. Strategy:BoD
32° x°
Note: The figure is not drawn to scale.

140. Strategy:BoD (#2)
The trapezoid shown below has a height of 12. Find the length of the base not given. 20 17
Note: The figure is not drawn to scale. 13

141. Practice
Work with your neighbor to complete the 6 practice problems. Try to use some of the strategies presented today to help you.
You have 12 minutes starting now.

142. On your mark, get set.....

143. START!

144. 12 minutes remaining

145. 10 minutes remaining

146. 5 minutes remaining

147. 2 minutes remaining

148. 1 minutes remaining

149. Time’s Up!!!!

150. Example 1:
In the figure, l m, and x is 20° less than y. What is the value of y?
A) 35 B) 45 C) 55 D) E) 100 l y° x° m

151. Example 2:
In the figure,if ∆ABC is the same size and shape as ∆ABD, then the degree measure of <BAD is ___?
A) 25 B) 35 C) 45 D) E) 75 B D
40°
70° E A C

152. Example 3:
In right triangle ABC, if the measure of <ABD = 15° ands <A = 30°, what is the length of DB?
A) 6 B) 6√3 C) 6√2 D) 6√3 - 6 E) 6√2 - 6 A
30° 12 D
15° B C E

153. Example 4:
If the lengths of two sides of a triangle are 14 and 23, then the perimeter :
I. must be between 9 and 37
II. must be between 46 and 74
III. must be greater than 50
A) I only B) I & II only C) I, II, & III D) II only E)None of the above

154. Example 5:
What is the area of a circle with a circumeference of π2?

155. Example 6:
Cube A has an edge of 4. If each edge of cube A is increased by 25%, creating a second cube B, then the volume of cube B is how much greater than the volume of cube A?
A) 16 B) C) 61 D) 64 E) 80

156. Be sure to turn this in to your math teacher the next time you go to math class!

157. Closing Comments

158. Today we will

159. Vocabulary Terms

160. GEOMETRY PRACTICE
Find the value of x:
A 37 B 47 C 57
D 90 E 133

161. GEOMETRYPRACTICE
Find the value of x:
A 23 B 33 C 43 D57 E 90

162. GEOMETRY PRACTICE
FIND THE VALUE OF X x

163. GEOMETRY PRACTICE
The complement of an angle is ___ more than the angle. What is the sum of the angle’s complement and its supplement?

164. Example 1:
In the figure, l m, and x is 20° less than y. What is the value of y?
A) 35 B) 45 C) 55 D) E) 100 l y° x° m

165. Example 2:
In the figure,if ∆ABC is the same size and shape as ∆ABD, then the degree measure of <BAD is ___?
A) 25 B) 35 C) 45 D) E) 75 B D
40°
70° E A C

166. Example 3:
In right triangle ABC, if the measure of <ABD = 15° ands <A = 30°, what is the length of DB?
A) 6 B) 6√3 C) 6√2 D) 6√3 - 6 E) 6√2 - 6 A
30° 12 D
15° B C E

167. Example 4:
If the lengths of two sides of a triangle are 14 and 23, then the perimeter :
I. must be between 9 and 37
II. must be between 46 and 74
III. must be greater than 50
A) I only B) I & II only C) I, II, & III D) II only E)None of the above

168. Example 5:
What is the area of a circle with a circumeference of π2?

169. Example 6:
Cube A has an edge of 4. If each edge of cube A is increased by 25%, creating a second cube B, then the volume of cube B is how much greater than the volume of cube A?
A) 16 B) C) 61 D) 64 E) 80