Geometry Cliff Notes

1
Geometry Cliff Notes

• Chapters 4 and 5

2

• Chapter 4
• Reasoning and Proof,
• Lines, and
• Congruent Triangles

3
Distance Formula

• d
• Example
• Find the distance between (3,8)(5,2)
• 
  d

4
Midpoint Formula

• M
• Example
• Find the midpoint (20,5)(30,-5)
• M

5
Conjecture

• An unproven statement that is based on
  observations.

6
Inductive Reasoning

• Used when you find a pattern in specific cases
  and then write a conjecture for the general case.

7
Counterexample

• A specific case for which a conjecture is false.
• Conjecture All odd numbers are prime.
• Counterexample The number 9 is odd but it is a
  composite number, not a prime number.

8
Conditional Statement

• A logical statement that has two parts,
• a hypothesis and a conclusion.
• Example All sharks have a boneless skeleton.
• Hypothesis All sharks
• Conclusion A boneless skeleton

9
If-Then Form

• A conditional statement rewritten. If part
  contains the hypothesis and the then part
  contains the conclusion.
• Original All sharks have a boneless skeleton.
• If-then If a fish is a shark, then it has a
  boneless skeleton.
• When you rewrite in if-then form, you may need
  to reword the hypothesis and conclusion.

10
Negation

• Opposite of the original statement.
• Original All sharks have a boneless skeleton.
• Negation Sharks do not have a boneless skeleton.

11
Converse

• To write a converse, switch the hypothesis and
  conclusion of the conditional statement.
• Original Basketball players are athletes.
• If-then If you are a basketball player, then you
  are
• an athlete.
• Converse If you are an athlete, then you are a
• basketball player.

12
Inverse

• To write the inverse, negate both the hypothesis
  and conclusion.
• Original Basketball players are athletes.
• If-then If you are a basketball player, then you
  are
• an athlete. (True)
• Converse If you are an athlete, then you are a
• basketball player. (False)
• Inverse If you are not a basketball player, then
  you
• are not an athlete. (False)

13
Contrapositive

• To write the contrapositive, first write the
  converse and then negate both the hypothesis and
  conclusion.
• Original Basketball players are athletes.
• If-then If you are a basketball player, then you
  are
• an athlete. (True)
• Converse If you are an athlete, then you are a
• basketball player. (False)
• Inverse If you are not a basketball player, then
  you
• are not an athlete. (False)
• Contrapositive If you are not an athlete, then
  you are not a
• basketball player.
  (True)

14
Equivalent Statement

• When two statements are both true or both false.

15
Perpendicular Lines

• Two lines that intersect to form a right angle.
• Symbol

16
Biconditional Statement

• When a statement and its converse are both true,
  you can write them as a single biconditional
  statement.
• A statement that contains the phrase if and only
  if.
• Original If a polygon is equilateral, then all
  of its
• sides are congruent.
• Converse If all of the sides are congruent, then
  it is
• an equilateral polygon.
• Biconditional Statement A polygon is equilateral
  if and only if
• all of
  its sides are congruent.

17
Deductive Reasoning

• Uses facts, definitions, accepted properties, and
  the laws of logic to form a logical statement.

18
Law of Detachment

• If the hypothesis of a true conditional statement
  is true, then the conclusion is also true.
• Original If an angle measures less than 90,
• then it is not obtuse.
• m ltABC 80
• ltABC is not obtuse

19
Law of Syllogism

• If hypothesis p, then conclusion q.
• If hypothesis q, then conclusion r.
• (If both statements above are true).
• If hypothesis p, then conclusion r
• Original If the power is off, then the fridge
  does
• not run. If the fridge does not
  run, then
• the food will spoil.
• Conditional Statement If the power if off, then
  the
• food will
  spoil.

20
Postulate

• A rule that is accepted without proof.

21
Theorem

• A statement that can be proven.

22
Subtraction Property of Equality

• Subtract a value from both sides of an equation.
• x 7 10
• -7 -7
• X 3

23
Addition Property of Equality

• Add a value to both sides of an equation.
• X-7 10
• 7 7
• X 17

24
Division Property of Equality

• Divide both sides by a value.
• 3x 9
• 3
• x 3

25
Multiplication Property of Equality

• Multiply both sides by a value.
• ½x 7
• 2 2
• x 14

26
Distributive Property

• To multiply out the parts of an expression.
• 2(x-7)
• 2x - 14

27
Substitution Property of Equality

• Replacing one expression with an equivalent
  expression.
• AB 12, CD 12
• AB CD

28
Proof

• Logical argument that shows a statement is true.

29
Two-column Proof

• Numbered statements and corresponding reasons
  that show an argument in a logical order.

30
Reflexive Property of Equality

• Segment
• For any segment AB, AB AB or AB AB
• Angle
• For any angle or

31
Symmetric Property of Equality

• Segment
• If AB CD then CD AB or AB CD
• Angle

32
Transitive Property of Equality

• Segment
• If AB CD and CD EF, then AB EF or ABEF
• Angle

33
Supplementary Angles

• Two Angles are Supplementary if they add up to
  180 degrees.

34
Complementary Angles

• Two Angles are Complementary if they add up to 90
  degrees
• (a Right Angle).

35
Segment Addition Postulate

• If B is between A and C, then AB BC AC.
• If AB BC AC, then B is between A and C.
• . . .
• A B C

36
Angle Addition Postulate

• If S is in the interior of angle PQR,
• then the measure of angle PQR is equal to
• the sum of the measures of angle PQS and
• angle SQR.

37
Right Angles Congruence Theorem

• All right angles are congruent.

38
Vertical AnglesCongruence Theorem

• Vertical angles are congruent.

39
Linear Pair Postulate

• Two adjacent angles whose common sides are
  opposite rays.
• If two angles form a linear pair, then they are
  supplementary.

40
Theorem 4.7

• If two lines intersect to form a linear pair of
  congruent angles, then the lines are
  perpendicular.

41
Theorem 4.8

• If two lines are perpendicular, then they
  intersect to form four right angles.

42
Theorem 4.9

• If two sides of two adjacent acute angles are
  perpendicular, then the angles are complementary.

43
Transversal

• A line that intersects two or more coplanar
  lines at different points.

44
Theorem 4.10

• Perpendicular Transversal Theorem
• If a transversal is perpendicular to one of two
  parallel lines, then it is perpendicular to the
  other.

45
Theorem 4.11

• Lines Perpendicular to a Transversal Theorem
• In a plane, if two lines are perpendicular to
  the same line, then they are parallel to each
  other.

46
Distance from a point to a line

• The length of the perpendicular segment from the
  point to the line.

• Find the slope of the line
• Use the negative reciprocal slope starting at
• the given point until you hit the line
• Use that intersecting point as your second
• point.
• Use the distance formula

47
Congruent Figures

• All the parts of one figure are congruent to the
  corresponding parts of another figure.
• (Same size, same shape)

48
Corresponding Parts

• The angles, sides, and vertices that are in the
  same location in congruent figures.

49
Coordinate Proof

• Involves placing geometric figures in a
  coordinate plane.

50
Side-Side-Side CongruencePostulate (SSS)

• If three sides of one triangle are congruent to
  three sides of a second triangle, then the two
  triangles are congruent.

51
Legs

• In a right triangle, the sides adjacent to the
  right angle are called the legs. (a and b)

52
Hypotenuse

• The side opposite the right angle. (c)

53
Side-Angle-Side CongruencePostulate (SAS)

• If two sides and the included angle of one
  triangle are congruent to two sides and the
  included angle of a second triangle, then the two
  triangles are congruent.

54
Theorem 4.12

• Hypotenuse-Leg Congruence Theorem
• If the hypotenuse and a leg of a right triangle
  are congruent to the hypotenuse and a leg of a
  second triangle, then the two triangles are
  congruent.

55
Flow Proof

• Uses arrows to show the flow of a logical
  statement.

56
Angle-Side-Angle Congruence Postulate (ASA)

• If two angles and the included side of one
  triangle are congruent to two angles and the
  included side of a second triangle, then the two
  triangles are congruent.

57
Theorem 4.13

• Angle-Angle-Side Congruence Theorem
• If two angles and a non-included side of one
  triangle are congruent to two angles and the
  corresponding non-included side of a second
  triangle, then the two triangles are congruent.

58
Chapter 5Relationships in Triangles
andQuadrilaterals
59
Midsegment of a Triangle

• Segment that connects the midpoints of two sides
  of the triangle.

60
Theorem 5.1

• Midsegment Theorem
• The segment connecting the midpoints of two sides
  of a triangle is parallel to the third side and
  is half as long as that side.
• x 3

61
Perpendicular Bisector

• A segment, ray, line, or plane that is
  perpendicular to a segment at its midpoint.

62
Equidistant

• A point is the same distance from each of two
  figures.

63
Theorem 5.2

• Perpendicular Bisector Theorem
• In a plane, if a point is on the perpendicular
  bisector of a segment, then it is equidistant
  from the endpoints of the segment.

64
Theorem 5.3

• Converse of the Perpendicular Bisector Theorem
• In a plane, if a point is equidistant from the
  endpoints of a segment, then it is on the
  perpendicular bisector of the segment.

65
Concurrent

• When three or more lines, rays, or segments
  intersect in the same point.

66
Theorem 5.4

• Concurrency of Perpendicular Bisectors of a
  Triangle
• The perpendicular bisectors of a triangle
  intersect at a point that is equidistant from the
  vertices of the triangle.

67
Circumcenter

• The point of concurrency of the three
  perpendicular bisectors of a triangle.

68
Angle Bisector

• A ray that divides an angle into two congruent
  adjacent angles.

69
Incenter

• Point of concurrency of the three angle bisectors
  of a triangle.

70
Theorem 5.5

• Angle Bisector Theorem
• If a point is on the bisector of an angle, then
  it is equidistant from the two sides of the
  angle.

71
Theorem 5.7

• Concurrency of Angle Bisectors of a Triangle
• The angle bisectors of a triangle intersect at a
  point that is equidistant from the sides of the
  triangle.

72
Theorem 5.6

• Converse of the Angle Bisector Theorem
• If a point is in the interior of an angle and is
  equidistant from the sides of the angle, then it
  lies on the bisector of the angle.

73
Median of a Triangle

• Segment from a vertex to the midpoint of the
  opposite side.

74
Centroid

• Point of concurrency of the three medians of a
  triangle. Always on the inside of the triangle.

75
Altitude of a Triangle

• Perpendicular segment from a vertex to the
  opposite side or to the line that contains the
  opposite side.

76
Orthocenter

• Point at which the lines containing the three
  altitudes of a triangle intersect.

77
Theorem 5.8

• Concurrency of Medians of a Triangle
• The medians of a triangle intersect at a point
  that is two thirds of the distance from each
  vertex to the midpoint of the opposite side.

78
Theorem 5.9

• Concurrency of Altitudes of a Triangle
• The lines containing the altitudes of a triangle
  are concurrent.

79
Theorem 5.10

• If one side of a triangle is longer than another
  side, then the angle opposite the longer side is
  larger than the angle opposite the shorter side.

80
Theorem 5.11

• If one angle of a triangle is larger than another
  angle, then the side opposite the larger angle is
  longer than the side
• opposite the smaller angle.

81
Theorem 5.12

• Triangle Inequality Theorem
• The sum of the lengths of the two smaller sides
  of a triangle must be greater than the length of
  the third side.

82
Theorem 5.13

• Exterior Angle Inequality Theorem
• The measure of an exterior angle of a triangle is
  greater than the measure of either of the
  nonadjacent interior angles.

83
Theorem 5.14

• HingeTheorem
• If two sides of one triangle are congruent to two
  sides of another triangle, and the included angle
  of the first is larger than the included angle of
  the second, then the third side of the first is
  longer than the third side of the second.

11cm
72
68
84
Theorem 5.15

• Converse of the HingeTheorem
• If two sides of one triangle are congruent to two
  sides of another triangle, and the third side of
  the first is longer than the third side of the
  second, then the included angle of the first is
  longer than the third side of the second.

11cm
72
68
85
Indirect Proof

• A proof in which you prove that a statement is
  true by first assuming that its opposite is true.
  If this assumption leads to an impossibility,
  then you have proved that the original statement
  is true.
• Example Prove a triangle cannot have 2 right
  angles.1) Given ?ABC.2) Assume angle A and
  angle B are both right angles is true by one of
  two possibilities (it is either true or false so
  we assume it is true).3) measure of angle A 90
  degrees and measure of angle B 90 degrees by
  definition of right angles.4) measure of angle A
  measure of angle B measure of angle C 180
  degrees by the sum of the angles of a triangle is
  180 degrees.5) 90 90 measure of angle C
  180 by substitution.6) measure of angle C 0
  degrees by subtraction postulate7) angle A and
  angle B are both right angles is false by
  contradiction (an angle of a triangle cannot
  equal zero degrees)8) A triangle cannot have 2
  right angles by elimination (we showed since that
  if they were both right angles, the third angle
  would be zero degrees and this is a contradiction
  so therefore our assumption was false ).

86
Diagonal of a Polygon

• Segment that joins two nonconsecutive vertices.

87
Theorem 5.16

• Polygon Interior Angles Theorem
• The sum of the measures of the interior angles of
  a polygon is 180(n-2).
• n number of sides

88
Corollary to Theorem 5.16

• Interior Angles of a Quadrilateral
• The sum of the measures of the interior angles of
  a quadrilateral is 360.

89
Theorem 5.17

• Polygon Exterior Angles Theorem
• The sum of the measures of the exterior angles of
  a convex polygon, one angle at each vertex, is
  360.

90
Interior Angles of the Polygon

• Original angles of a polygon. In a regular
  polygon, the interior angles are congruent.

91
Exterior Angles of the Polygon

• Angles that are adjacent to the interior angles
  of a polygon.

92
Parallelogram

• A quadrilateral with both pairs of opposite sides
  parallel.

93
Theorem 5.18

• If a quadrilateral is a parallelogram, then its
  opposite sides are congruent.

94
Theorem 5.19

• If a quadrilateral is a parallelogram, then its
  opposite angles are congruent.

95
Theorem 5.20

• If a quadrilateral is a parallelogram, then its
  consecutive angles are supplementary.

96
Theorem 5.21

• If a quadrilateral is a parallelogram, then its
  diagonals bisect each other.

97
Theorem 5.22

• If both pairs of opposite sides of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

98
Theorem 5.23

• If both pairs of opposite angles of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

99
Theorem 5.24

• If one pair of opposite sides of a quadrilateral
  are congruent and parallel, then the
  quadrilateral is a parallelogram.

100
Theorem 5.25

• If the diagonals of a quadrilateral bisect each
  other, then the quadrilateral is a parallelogram.

101
Rhombus

• A parallelogram with four congruent sides.

102
Rectangle

• A parallelogram with four right angles.

103
Square

• A parallelogram with four congruent sides and
  four right angles.

104
Rhombus Corollary

• A quadrilateral is a Rhombus if and only if it
  has four congruent sides.

105
Rectangle Corollary

• A quadrilateral is a Rectangle if and only if it
  has four right angles.

106
Square Corollary

• A quadrilateral is a Square if and only if it is
  a Rhombus and a Rectangle.

107
Theorem 5.26

• A parallelogram is a Rhombus if and only if its
  diagonals are
• perpendicular.

108
Theorem 5.27

• A parallelogram is a Rhombus if and only if each
  diagonal bisects a pair of opposite angles.

109
Theorem 5.28

• A parallelogram is a Rectangle if and only if its
  diagonals are congruent.

110
Trapezoid

• A quadrilateral with exactly one pair of parallel
  sides.

111
Base of a Trapezoid

• Parallel sides of a trapezoid.

112
Legs of a Trapezoid

• Nonparallel sides of a trapezoid.

113
Isosceles Trapezoid

• Trapezoid with congruent legs.

114
Midsegment of a Trapezoid

• Segment that connects the midpoints of its legs.

115
Kite

• A Quadrilateral that has two pairs of consecutive
  congruent sides, but opposite sides are NOT
  congruent.

116
Theorem 5.29

• If a Trapezoid is Isosceles, then each pair of
  base angles is congruent.

117
Theorem 5.30

• If a Trapezoid has a pair of congruent base
  angles, then it is an Isosceles Trapezoid.

118
Theorem 5.31

• A Trapezoid is Isosceles if and only if its
  diagonals are congruent.

119
Theorem 5.32

• Midsegment Theorem for Trapezoids
• The midsegment of a trapezoid is parallel to each
  base and its length is one half the sum of the
  lengths of the bases.

120
Theorem 5.33

• If a quadrilateral is a kite, then its diagonals
  are perpendicular.

121
Theorem 5.34

• If a quadrilateral is a kite, then exactly one
  pair of opposite angles are congruent.