1. Day 1-2: Objectives 10-3 & 4-7
To define and identify the Incenter, Circumcenter, Orthocenter and Centroid of triangles.
To apply the definitions of the median and altitude of a triangle and perpendicular bisector of a segment.
To state and apply the theorem about a point on the perpendicular bisector of a segment, and the converse.
To state and apply the theorem about a point on the bisector of an angle, and the converse.

2. Median: Is a segment from a vertex to the midpoint of the opposite side. Centroid: Is the point where the medians of a triangle intersect.

3. Median: Is a segment from a vertex to the midpoint of the opposite side. Centroid: Is the point where the medians of a triangle intersect.

4. Median: Is a segment from a vertex to the midpoint of the opposite side. Centroid: Is the point where the medians of a triangle intersect.

5. Theorem 10-4: The medians of a triangle intersect in a point called the Centroid, that is two thirds of the distance from each vertex to the midpoint of the opposite side.

6. Altitude: Is a perpendicular segment from a vertex to the line that contains the opposite side. Theorem 10-3: The lines that contain the altitudes of a triangle intersect in a point called the Orthocenter.

7. Altitude: Is a perpendicular segment from a vertex to the line that contains the opposite side. Theorem 10-3: The lines that contain the altitudes of a triangle intersect in a point called the Orthocenter.

8. Altitude: Is a perpendicular segment from a vertex to the line that contains the opposite side. Theorem 10-3: The lines that contain the altitudes of a triangle intersect in a point called the Orthocenter.

9. Perpendicular bisector of a segment: Is a line, ray, or segment that is perpendicular to the segment at its midpoint. This line is unique.

10. Theorem 10-2: The perpendicular bisectors of the sides of a triangle intersect in a point called the Circumcenter, that is equidistant from the three vertices of the triangle.

11. Theorem 10-2: The perpendicular bisectors of the sides of a triangle intersect in a point called the Circumcenter, that is equidistant from the three vertices of the triangle.

12. Theorem 10-2: The perpendicular bisectors of the sides of a triangle intersect in a point called the Circumcenter, that is equidistant from the three vertices of the triangle.

13. Theorem 4-5: If a point lies on the perpendicular bisector of a segment, then the point is equidistant from the endpoints of the segment.

14. Theorem 4-6: If a point is equidistant from the endpoints of the segment, then the point lies on the perpendicular bisector of the segment.

15. Distance from a point to a line or plane: Is the length of the perpendicular segment from the point to the line or plane.

16. Theorem 10-1: The bisectors of the angles of a triangle intersect in a point called the Incenter, that is equidistant from the three sides of the triangle.

17. Theorem 10-1: The bisectors of the angles of a triangle intersect in a point called the Incenter, that is equidistant from the three sides of the triangle.

18. Theorem 10-1: The bisectors of the angles of a triangle intersect in a point called the Incenter, that is equidistant from the three sides of the triangle.

19. Theorem 4-7: If a point lies on the bisector of an angle, then the point is equidistant from the sides of the angle.

20. Theorem 4-8: If a point is equidistant from the sides of an angle, then the point lies on the bisector of an angle.

21. Example 1
Centroid

22. Example 2
Incenter

23. Example 3
Orthocenter

24. Example 4
Circumcenter

25. Example 5 Points D, E, F are midpoints of the
sides of the triangle. What is the
name of point G?
Centroid

26. Example 6
Circumcenter

27. Example 7
Orthocenter

28. Example 8
Incenter

29. Example 9
Incenter

30. Example 10 D,E,F are midpoints of the sides of the triangle.
What is the name of point G?
Centroid

31. Example 11
Circumcenter

32. Example 12
Orthocenter

33. 10 4 3 6

34. Median
Altitude
SAS
Isosceles S T RS RT

36. Assignment WS 1

37. Objectives Section 5-3:
To apply theorems about parallel lines and the segment that joins the midpoints of two sides of a triangle.

38. Theorem 5-8: If two lines are parallel, then all points on one line are equidistant from the other line. A B l m C D

39. Theorem 5-9: If three parallel lines cut off congruent segments on one transversal, then they cut off congruent segments on every transversal. A B C X Y Z R S
Theorem 5-8: If two lines are parallel, then all points on one
line are equidistant from the other line.

40. Theorem 5-10: A line that contains the midpoint of one side of a triangle and is parallel to another side passes through the midpoint of the third side. A B C l M N

41. Theorem 5-11: The segment that joins the midpoints of two sides of a triangle (Mid-segment) is,
Parallel to the third side.
Half as long as the third side. A B C M N 10 5

42. 5 14 8

43. 5 7 4 5 7 4 5 7 4 5 7 4

44. Assignment WS 2 Page

45. Objectives Section 6-1:
To apply properties of inequality to positive numbers, lengths of segments, and measures of angles.

46. “A property of inequality”.
If one of the following properties of inequality are used in a proof, write for the reason,
“A property of inequality”.

47. Class Exercises: Write <, = or > for each of the following.

48. Class Exercises: Write <, = or > for each of the following.70

49. A B C E D
Statements
Reasons

50. F E D 1
Statements
Reasons

51. Theorem 6-1: The Exterior Angle Inequality Theorem: The measure of an exterior angle of a triangle is greater than the measure of either remote interior angle.

52. Class Exercises: 11. 1 2 3 4
Statements
Reasons

53. Assignment WS 3

54. Objectives Section 6-4:
To state and apply the inequality theorems and corollaries for one triangle.

55. Theorem 6-2: If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.
Class Exercises: Name the largest and smallest angles of each triangle. 1. 2. 3.
Largest
Smallest
Largest
Smallest
Largest
Smallest

56. Class Exercises: Name the longest and shortest sides of each triangle.
Theorem 6-3: If one angle of a triangle is larger than a second angle, then the side opposite the first angle is longer than the side opposite the second angle.
Class Exercises: Name the longest and shortest sides of each triangle. 27 4. 5. 6. 47 35
Longest
Shortest
Longest
Shortest
Longest
Shortest

57. Corollary 1: The perpendicular segment from a point to a line is the shortest segment from the point to the line. Corollary 2: The perpendicular segment from a point to a plane is the shortest segment from the point to the plane.

58. 43 51
False
True
True

59. Theorem 6-4: The Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.
Class Exercises: Is it possible for a triangle to have sides with the given lengths?
10. 8, 9, 10 ______ , 6, 20 ______
12. 7, 7, 14.1 ______ , 11, 16 ______
, 0.6, 1 ______ , 18, 18 ______
Yes No
True
False
True
True No No
False
False
Yes
Yes
True
True
True
True
True

60. Class Exercises: 16. The lengths of two sides of a triangle are 3 and 5. The length of the third side must be greater than _____, but less than _____? 2 8

61. Assignment WS 4

62. Objectives Section 6-5:
To state and apply the inequality theorems for two triangles.

63. Theorem 6-5 SAS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the included angle of the first triangle is larger than the included angle of the second triangle, then the third side of the first triangle is longer than the third side of the second triangle.

64. Theorem 6-6 SSS Inequality Theorem: If two sides of one triangle are congruent to two sides of another triangle, and the third side of the first triangle is longer than the third side of the second triangle, then the included angle of the first triangle is larger than the included angle of the second triangle.

65. Class Exercises: What can be deduced? State the
reason that supports your answer. 1. C F 15 14 A B D E

66. Class Exercises: What can be deduced? State the
reason that supports your answer. 2.
SAS Ineq. Thm.

67. Class Exercises: What can be deduced? State the
reason that supports your answer. 3. T 1 2 R S V

68. Class Exercises: What can be deduced? State the
reason that supports your answer. 4. F C 1 2 B E A D

69. Class Exercises: What can be deduced? State the
reason that supports your answer. 5. U 7 40 N G 40 7 O

70. Class Exercises: What can be deduced? State the
reason that supports your answer. 6. T 31 32 n n A C B

71. Class Exercises: What can be deduced? State the
reason that supports your answer. 7. 10 1 5 6 2 10

72. Class Exercises: What can be deduced? State the
reason that supports your answer. 8. 88 92 F G 18 D E

73. Assignment WS 5 Page 231 1, 3-11.

74. Assignment WS 6R

75. Objectives Section 7-1 & 7-2:
To express a ratio in simplest form.
To solve for an unknown term in a given proportion.
To express a given proportion in an equivalent form.

76. Ratio: Is the quotient of two terms with the same unit of
measure and always written in simplest form.
Extended Ratio: Compares three or more terms.

77. Sometimes

79. No common unit of measure
Yes
Yes No
Yes

80. Proportion: Is an equation stating that two ratios are equal.
Extended Proportion: Is an equation stating that three or more ratios are equal.

81. Means-Extremes Property of Proportions: The product of the extremes equals to the product of the means.

82. Properties of Proportions

85. Assignment WS 7 Page 243 6-28, 30. Page 247 1-28, 34, 36, 38.

86. Objectives Sections 7-3, 7-4 & 7-5:
To state and apply the properties of similar polygons.
To use the AA Similarity Postulate, the SAS Similarity Theorem, and the SSS Similarity Theorem to prove triangles are similar.
To use similar triangles to deduce information about segments or angles.

87. Properties of Similarity
Properties of Similarity
Reflexive
Symmetric
Transitive

88. Are the polygons similar?

89. Scale Factor: Is a ratio of the lengths of two corresponding sides of similar polygons.
The ratio of the perimeters of similar polygons is the same
as the scale factor.

91. No. Corr. sides aren’t proportional.
Yes.
No. Corr. angles aren’t congruent.
No. Corr. angles aren’t congruent.

92. No. Corr. angles aren’t congruent. Yes.
Yes.
No. Corr. angles aren’t congruent.
Yes. 10 20 F G H 15 R S X 16 24 32 70 C D E 40 60 N U J 70
Shortest
Remaining
Longest 12 R S Q 8 6 T U 9

93. No, angles aren’t included Yes
No, angles aren’t included
Yes 12 W X V 8 6 Y Z 9 20 16 24 A C D B 36 30
Shortest
Remaining
Longest

94. Directions: Complete.
10. If the corresponding angles of two polygons are congruent, must the polygons be similar? _____
11. If the corresponding sides of two polygons are in proportion, must the polygons be similar? ____
12. Two polygons are similar. Do they have to be congruent? ___
13. Two polygons are congruent. Do they have to be similar? __ No No No
Yes

95. Corr. Angles are not congruent

96. No
Yes No No
Yes No
Yes
Yes

98. Assignment WS 8 Page 250 1-4, 11-14, 26. Page 257 1-7, 10-17, 21-23
Assignment WS 8 Page , 11-14, 26. Page , 10-17, Page

99. Objectives Section 7-6:
To state and apply the Triangle Proportionality Theorem and its corollary.
To state and apply the Triangle Angle-Bisector Theorem.

100. Triangle Proportionality Theorem: If a line parallel to one side of a triangle intersects the other two sides, then it divides those sides proportionally. a b d c
Corollary: If three parallel lines intersect two transversals, then they divide the transversals proportionally.

101. Triangle Angle-Bisector Theorem: If a ray bisects an angle of a triangle, then it divides the opposite side into segments proportional to the other two sides. d c a b

104. Assignment WS 9 Page ,

105. Objectives Section 8-1:
To determine the geometric mean between two numbers.
To state and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle.

108. Theorem 8-1: If the altitude is drawn to the hypotenuse of a right triangle, then the two triangles formed are similar to the original triangle and to each other.
Corollary 1: When the altitude is drawn to the hypotenuse of a right triangle, the length of the altitude is the geometric mean between the segments of the hypotenuse.
Corollary 2: When the altitude is drawn to the hypotenuse of a right triangle, each leg is the geometric mean between the hypotenuse and the segment of the hypotenuse that is adjacent to that leg.

110. Assignment WS 11 Page 288 16-39 odd.

111. Objectives Sections 8-2 & 8-3:
To state and apply the Pythagorean Theorem.
To state and apply the converse Pythagorean Theorem and related theorems about obtuse and acute triangles.

112. Pythagorean Theorem: In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.

114. Theorem 8. 3 (Converse of the Pythagorean Thm
Theorem 8.3 (Converse of the Pythagorean Thm.): If the square of one side of a triangle is equal to the sum of the squares of the other two sides, then the triangle is a right triangle. c A B C b a

118. Assignment WS 12 Page 292 1-8, 12, 20-22. Page 297 1-3, 9-15

119. Objectives Section 8-4:
To determine the lengths of two sides of a 45o-45o-90o or 30o-60o-90o triangle when the length of the third side is known.

120. c a
45o
60o
30o B C A c a b

124. Assignment WS 13 Page

125. SAS Similarity TheoremD B E C F X Y 1

126. SSS Similarity TheoremD B E C F X Y 1

127. 1 2 F D G E K 3 4

128. Example S Z W T R

129. Example S Z W T R 1 2

130. Example S Z W T R

131. T P R S Q 1 3 2 4
Statements
Reasons

132. Corollary If three parallel lines intersect two transversals, then
they divide the transversals proportionally. R S T X Y Z N

133. Example D P E R Q F 12 8 10 6 4 5

134. Example U T S D R C A B

135. Theorem 6-2: If one side of a triangle is longer than a second side, then the angle opposite the first side is larger than the angle opposite the second side.

136. Class Exercises: Find the ratio of BC to AC.
Class Exercises: Find the ratio of the measure of the smallest angle of the triangle to that of the largest angle. 6b 14 B A C 60 70 50

137. Class Exercises: A poster is 1 m long and 52 cm wide
Class Exercises: A poster is 1 m long and 52 cm wide. Find the ratio of the width to the length.

138. Class Exercises: The measures of the three angles of a triangle are in the ratio 2 : 2 : 5. Find the measure of each angle.

139. Class Exercises: Solve.

140. Class Exercises: Solve.F O I L

141. Class Exercises: Solve.F O I L

142. Classroom Exercises

143. Classroom Exercises

144. Classroom Exercises

145. Classroom Exercises

146. Example A B D E C

147. Example A B D E C

148. Similar Triangles?

149. Similar Triangles?

150. Similar Triangles?

151. Similar Triangles?

152. Similar Triangles?

153. Solve.

154. 1 2 K H O G F
Example
Statements
Reasons

155. Theorem 8-6: 45o-45o-90o Theoremc a

156. Theorem 8-7: 30o-60o-90o TheoremB C A c a
60o
30o b B C D A