1. Chapter 1

2. Vocab
A net is a two-dimensional diagram that you can fold to form a three-dimensional figure.

3. Vocab Terms:
POINT- has no dimension (no length, width, or thickness) represented by a dot
Point C

4. Vocab Terms:

5. Vocab Terms:
BCD

6. Vocab Terms:

7. Vocab Terms:

8. Postulate 1-1

9. Postulate 1-2

10. Postulate 1-3

11. Postulate 1-4

13. Congruent Angles - angles that have the same measure.

15. Types of Angles:
Two angles are adjacent angles if they share a common vertex and side, but have no common interior points.

16. Angles 1 and 3 are vertical angles, so are angles 2 and 45 4 2 6 3
Vertical angles – two angles whose sides form two pairs of opposite rays. Vertical angles are congruent.
Angles 1 and 3 are vertical angles, so are angles 2 and 4
Linear pair – two adjacent angles whose uncommon sides are opposite rays.
Angles 5 and 6 are a linear pair and add up to 180°.

17. Complementary Angles:
Two angles whose sum is 90°.
Note that these two angles
can be “pasted” together
to form a right angle.

18. Supplementary Angles:
Two angles whose sum is 180°.
Note that these two angles
can be “pasted” together
to form a straight angle.

19. An angle bisector is a ray that divides an angle into two congruent angles. Its endpoint is at the angle vertex.

20. Midpoint
The midpoint of a segment is the point that divides, or bisects, the segment into two congruent segments.
Midpoint Formula:
Find the midpoint between
(-5,-3) and (1,-7).

21. Distance Formula
-computes the distance between two points in a coordinate plane

22. Classifying Polygons
Polygon – a closed plane figure formed by three or more segments. Each segment is called a side. Each endpoint of a side is called a vertex.
Each segment intersects exactly two other segments at their endpoints.
No two segments with a common endpoint are collinear.

23. Classifying Polygons
You can classify a polygon by the number of sides.

24. Classifying Polygons
A diagonal is a segment that connects two nonconsecutive vertices.

25. Circle r
Tell students that we are now going to explore the different formulas to find perimeter, area, and circumference of different shapes. Please highlight that these are very important and they should definitely write them in their notes as well as what each letter represents.
A circle uses circumference instead of perimeter to measure the outside length of the shape.
r represents the radius of the circle
d represents the diameter
The area of a circle formula uses units squared!
units
units2

26. Square P = 4s units A = s2 units2 s
Perimeter is 4 times the length of a side.
Area is the length of a side squared.

27. Rectangle P = 2l + 2w units A = lw units2 l w
Perimeter is length + width + length + width or (2)length + (2)width.
Area is length times width.
P = 2l + 2w units
A = lw units2 w

28. Triangle P = a + b + c units A = ½bh units2 c a h b
Perimeter adds up all the outside edges.
Area is ½ the base of the triangle times the height. b

29. Chapter 2

30. Using Inductive Reasoning
In the previous examples you used inductive reasoning to make a conjecture.
Conjecture: an educated guess about what you think is true based on observations.
What conjecture can you make about the twenty-first term in R, W, B, R, W, B, ...?

31. Finding a Counterexample
Not all conjectures turn out to be true. You can prove that a conjecture is false by finding at least one counterexample.

32. Conditional Statements
Type of logical statement
Has 2 parts- a hypothesis and a conclusion
Can be written in “if-then” form- the “if” part contains the hypothesis and the “then” part contains the conclusion.

33. Truth Value The truth value of a conditional is either true or false.
To show it is true, every time the hypothesis is true, the conclusion must also be true.
Example: If you live in the United States, then you live in North America.
To show it is false, find only one case where the hypothesis is true and the conclusion is false.
Example: If you live in North America, then you live in the United States.

34. Negation The negation of a statement is the opposite of the statement.
The symbol “~p” is read “not p”
Example:
Statement: “The ocean is green.”
Negation: “The ocean is not green.”

35. Related Conditional Statements

36. Equivalent Statements: when two statements are both true or both false.
ORIGINAL- If angle A is 30°, then angle A is acute.
INVERSE- If angle A is not 30°, then angle A is not acute.
CONVERSE- If angle A is acute, then angle A is 30°.
CONTRAPOSITIVE- If angle A is not acute, then angle A is not 30°.
BOTH
TRUE
BOTH
FALSE

37. Biconditional Statement:
A biconditional is a single true statement that combines a true conditional and its true converse.
A statement that contains the phrase “if and only if”.
Conditional: If it is Sunday, then I am watching football.
Converse: If I am watching football, then it is Sunday.
It is Sunday, if and only if I am watching football. (p q).

38. Biconditional Statements:
Can be either true or false.
To be true, BOTH the conditional and its converse must be true.
A TRUE Biconditional Statement is true both “forward” and “backward”.
All definitions can be written as a biconditional statements.

39. Identifying a Good Definition
A good definition is a statement that can help you identify or classify an object.
Uses clearly understood terms.
Is precise. Avoid words such as large, sort of, and almost.
Is reversible. Can be written as a true biconditional.
**One way to show it is NOT a good definition is to find a counterexample.

40. Symbolic Notation
Conditional statement has a hypothesis and a conclusion.
Written with symbolic notation: p represents hypothesis, q represents conclusion.
If p, then q.
Symbolic notation: p q
p implies q

41. Converse?
If q, then p or
Biconditional?
p if and only if q or

42. Contrapositive? Inverse? If not q, then not p. or
If not p, then not q. or
Contrapositive?
If not q, then not p. or

43. Checkpoint:
Let p be “You are a pianist” and q be “you are a musician”.
Write in words:

44. Laws of Logic:
DEDUCTIVE REASONING – uses facts, definitions, and accepted properties in a logical order to write a logical argument.
INDUCTIVE REASONING – uses examples and patterns to form a conjecture.

45. 2 LAWS OF DEDUCTIVE REASONING
LAW OF DETACHMENT:
If is a true conditional
statement and p is true, then q is true.

46. 2ND LAW LAW OF SYLLOGISM: If and are true conditional statements, then
is true.

47. Algebraic Properties

48. Theorem – a conjecture or statement that you prove true
***When writing a proof for a theorem, separate the theorem into a hypothesis and conclusion. The hypothesis becomes the “Given” statement and the conclusion is what you want to prove.

49. Chapter 3

50. Skew Lines – lines that do not intersect are not coplanar.
Parallel Lines – two lines that are coplanar and do not intersect. Notation: //
Skew Lines – lines that do not intersect are not coplanar.
Parallel Planes – two planes that do not intersect.

51. TRANSVERSAL – a line that intersects two or more coplanar lines at different points.1 2 3 4 5 6 7 8
Notice, angles 3, 4, 5, and 6 are interior angles (between the lines). Angles 1, 2, 7, and 8 are exterior angles (outside the lines).

52. CORRESPONDING ANGLES: occupy corresponding positions1 2 3 4 6 5 7 8
Corresponding angles lie on the same side of the transversal, and in corresponding positions

53. ALTERNATE EXTERIOR ANGLES: lie outside the two lines on opposite sides of the transversal.1 2 3 4 5 6 7 8

54. ALTERNATE INTERIOR ANGLES: lie between the two lines on opposite sides of the transversal.1 2 3 4 5 6 7 8

55. SAME-SIDE INTERIOR ANGLES: lie between the two lines on the same side of the transversal.3 5 1 2 4 6 7 8

56. Theorem 3-10: Triangle Sum Theorem
The sum of the measures of the interior angles of a
triangle is 180°.

57. Theorem 3-11: Exterior Angle Theorem
The measure of an exterior angle of a triangle equals
the sum of the measures of the two remote interior
angles.

58. SLOPE Definition Symbols Diagram
The slope, m, of a line is the ratio of the vertical change (rise) to the horizontal change (run) between any two points.
A line contains the points (x1, y1) and (x2, y2)

59. Equations of Lines: Y = mx + b SLOPE-INTERCEPT FORM slope
Y-intercept :the y-coordinate
of the point where the line
crosses the y-axis.
SLOPE-INTERCEPT FORM
*only used for nonvertical lines

60. Point-Slope Form
Standard Form

61. Checkpoint:
Find the slope of a line that passes through the points (-3, 0) and (4,7).

62. Checkpoint:
Find all three forms of the equation of the lines below. Line p, passes through (0, -3) and (1, -2). Line r passes through (-6, -1) and (3, 7).

63. Graphing Lines Identify the form of the equation.
Identify/Graph the y-intercept
Start at the y-intercept and use the slope to graph a couple points on the line
Connect the points

64. Using Two Points to Write an Equation
Write the equation of the line using the point(-2, -1) in point-slope form.
Find the slope of the line
Use the given values and plug into the point-slope form
Write the equation of the line in slope-intercept form.

65. Writing Equations of Horizontal and Vertical Lines

66. Slopes of Parallel Lines
When two line are parallel, their slopes are the same.
**If two lines are parallel, their slopes will be the same, but they MUST have a different y-intercept!
**Do all horizontal lines have the same slope? Explain.

67. Sage-n-Scribe

68. Slopes of Perpendicular Lines
If two lines are perpendicular, their slopes are negative reciprocals. In a coordinate plane, two nonvertical lines are perpendicular if and only if the product of their slopes is -1 (opposite reciprocals). Vertical and horizontal lines are perpendicular.

69. Perpendicular Slopes:
Opposite Reciprocals of each other.

70. Chapter 4

71. Congruent figures – have exactly the same size and shape
Congruent figures – have exactly the same size and shape. (Congruent Corresponding Parts)
*Corresponding angles
and sides are congruent.

72. Theorem 4.1: Third Angles Theorem
If two angles of one
triangle are congruent
to two angles of another
triangle, then the third
angles are also congruent.
If and
then

73. Triangles: figure formed by 3 segments joining 3 noncollinear points.
CLASSIFICATION BY SIDES
(# of congruent sides)
Scalene Triangle-no congruent sides
Isosceles Triangle-at least 2 congruent sides
Equilateral Triangle-3 congruent sides

74. CLASSIFICATION BY ANGLES
(all triangles have at least two acute angles, the third angle is used to classify)
EQUIANGULAR TRIANGLE – 3 congruent angles
(also acute)

75. Vertex – each of the three points joining the sides of a triangle.
Vertex A
Adjacent sides – two sides sharing a common vertex.
Sides AB and AC
Opposite side – the third side not sharing the vertex
Side BC

76. Postulate 4-1: Side-Side-Side (SSS) Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two triangles are congruent.

77. Postulate 4-2: Side-Angle-Side (SAS) Congruence Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of second triangle, then the two triangles are congruent.

78. Postulate 4-3: Angle-Side-Angle (ASA) Postulate
If 2 angles and the included side of one triangle are congruent to 2 angles and the included side of a second triangle, then the 2 triangles are congruent

79. Theorem 4-2: Angle-Angle-Side (AAS) Theorem
If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of the second triangle, then the two triangles are congruent.

80. Theorem 4.3 Isosceles Triangle Theorem
If two sides of a triangle are congruent, then the angles opposite them are congruent.
Theorem Converse:
If two angles of a triangle are congruent, then the sides opposite them are congruent

81. Right Triangles Hypotenuse:
It is the side opposite the right angle in a right triangle.
It is the longest side in a right triangle

82. AAS, SSS, SAS, ASA, and… For RIGHT triangles ONLY:
Theorem 4.6: Hypotenuse-Leg (HL) Congruence
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of a second right triangle, then the two triangles are congruent.

83. Chapter 5

84. Midsegment of a triangle is a segment that connects the midpoints of two sides of a triangle.

85. Theorem 5.1: Midsegment Theorem
If a segment joins the midpoints of two sides of a triangle, then the segment is parallel to the third side and is half as long. x
½ x

86. Perpendicular Bisector: A segment, ray, line, or plane that is perpendicular to a segment at its midpoint. C A B M D
Equidistant from two points – distance from each point is the same

87. Theorems:
Theorem 5-4: Angle Bisector Theorem If a point is on the bisector of an angle, then it is equidistant from the sides of the angle.
Theorem 5-5: Converse of the Angle Bisector Theorem If a point in the interior of an angle is equidistant from the sides of the angle, then it lies on the bisector of the angle. B D A C

88. Perpendicular Bisectors of a Triangle
A perpendicular bisector of a side of a triangle is a line (or ray or segment) that is perpendicular to a side of the triangle at the midpoint of the side.

89. Using Angle Bisectors of a Triangle
An angle bisector of a triangle is a bisector of an angle of a triangle.
The point of concurrency of the angle bisectors is called the incenter of the triangle. (Center of the inscribed circle.)It always lies inside the triangle.

90. Median of a Triangle
A median of a triangle is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. A
median B C
Every triangle has three medians.

91. Point of Concurrency for Medians
The point of concurrency for the 3 medians is called the centroid. The centroid is always on the inside of the triangle.
Centroid

92. Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is two thirds of the distance from each vertex to the midpoint of the opposite side.
BG = (2/3)BE, AG = (2/3)AF, and CG = (2/3)CD E D F
Note: The centroid of a triangle can be used as its balancing point.

93. Altitude of a Triangle
The altitude of a triangle is the perpendicular segment from a vertex to the opposite side or to the line that contains the opposite side.
An altitude can lie inside, on, or outside the triangle.

94. Orthocenter of a Triangle
Theorem 5.9: Concurrency of Altitudes Theorem:
The lines that contain the altitudes of a triangle are concurrent.
If segments AH, BH and CH are the altitudes of triangle DEF, the lines AH, BH, and CH intersect at a point H, the orthocenter. B A C

95. Summary

96. Writing an Indirect Proof
Step 1: State as a temporary assumption the opposite (negation) of what you want to prove Step 2: Show that this temporary assumption leads to a contradiction Step 3: Conclude that the temporary assumption must be false and that what you want to prove must be true

97. Triangle Inequality Theorem
Theorem. 5.12: The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

98. Hinge Theorem (SAS Inequality Theorem)
Theorem 5.13: If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle. V R
RT > VX 80
100 W X T S