Geometry Cliff Notes

Chapters 4 and 5

1

Chapter 4Reasoning and

Proof,Lines, and

Congruent Triangles2

Distance Formula

d=

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

d=

22 )82()35( 22 )6()2(

212

212 )()( yyxx

364

40

1023

Midpoint Formula

M=

Example:Find the midpoint (20,5)(30,-5)

M=

2,

22121 yyxx

2

55,

2

3020

)0,25(4

Conjecture

An unproven statement that is based on observations.

5

Inductive Reasoning

Used when you find a pattern in specific cases and then write a conjecture for the

general case.

6

CounterexampleA 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.

7

Conditional StatementA logical statement that has two parts,

a hypothesis and a conclusion.

Example: All sharks have a boneless skeleton.

Hypothesis: All sharksConclusion: A boneless skeleton

8

If-Then FormA 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.** 9

NegationOpposite of the original statement.

Original: All sharks have a boneless skeleton.

Negation: Sharks do not have a boneless skeleton.

10

ConverseTo 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.

11

InverseTo 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)

12

ContrapositiveTo 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) 13

Equivalent Statement

When two statements are both true or both false.

14

Perpendicular Lines

Two lines that intersect to form a right angle.

Symbol:

15

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.

16

Deductive Reasoning

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

statement.

17

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 <ABC = 80°

<ABC is not obtuse18

Law of SyllogismIf 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. 19

Postulate

A rule that is accepted without proof.

20

TheoremA statement that can be

proven.

21

Subtraction Property of Equality

Subtract a value from both sides of an equation.

x +7 = 10 -7 -7X = 3

22

Addition Property of Equality

Add a value to both sides of an equation.

X-7 = 10+7 +7X = 17

23

Division Property of Equality

Divide both sides by a value.

3x = 93 3x = 3

24

Multiplication Property of Equality

Multiply both sides by a value.

½x = 7·2 ·2 x = 14

25

Distributive Property

To multiply out the parts of an expression.

2(x-7)2x - 14

26

Substitution Property of Equality

Replacing one expression with an equivalent

expression.

AB = 12, CD = 12AB= CD

27

ProofLogical argument that

shows a statement is true.

28

Two-column ProofNumbered statements and

corresponding reasons that show an argument in a logical order.

# Statement Reason

1 3(2x-3)+1 = 2x Given2 6x - 9 + 1 = 2x Distributive Property3 4x - 9 + 1 = 0 Subtraction Property of Equality4 4x - 8 = 0 Add/Simplify5 4x = 8 Addition Property of Equality6 x = 2 Division Property of Equality

29

Reflexive Property of Equality

Segment:For any segment AB, AB AB or AB =

AB

Angle:For any angle or

,A A A A A

30

Symmetric Property of Equality

Segment:If AB CD then CD AB or AB =

CD

Angle: ,If A B then B A

31

Transitive Property of Equality

Segment: If AB CD and CD EF, then AB EF or

AB=EF

Angle:

,If A Band B C then A C

32

Supplementary Angles

Two Angles are Supplementary if they add up to 180

degrees.

33

Complementary AnglesTwo Angles are Complementary if

they add up to 90 degrees (a Right Angle).

34

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

35

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.

36

Right Angles Congruence Theorem

All right angles are congruent.

37

Vertical AnglesCongruence Theorem

Vertical angles are congruent.

1 3

2 4

or

38

Linear Pair PostulateTwo adjacent angles whose

common sides are opposite rays.

If two angles form a linear pair, then they are supplementary.

39

Theorem 4.7If two lines intersect to form a

linear pair of congruent angles, then the lines are

perpendicular.

ADB CDB

D

C

B

A

40

Theorem 4.8If two lines are perpendicular,

then they intersect to form four right angles.

41

Theorem 4.9If two sides of two adjacent acute

angles are perpendicular, then the angles are complementary.

a and b are complementary

ba

n

m

42

Transversal A line that intersects two or

more coplanar lines at different points.

c is the transversal

c

b

a

43

Theorem 4.10Perpendicular Transversal Theorem If a transversal is perpendicular to one of two parallel lines, then it is

perpendicular to the other.a b

c

b

a

44

Theorem 4.11Lines Perpendicular to a Transversal Theorem

In a plane, if two lines are perpendicular to the same line, then they are parallel

to each other.

c

b

a

45

Distance from a point to a line

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

m

A

•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

46

Congruent FiguresAll the parts of one figure are congruent to the corresponding

parts of another figure.

(Same size, same shape)

47

Corresponding PartsThe angles, sides, and vertices that are in the same location in congruent

figures.

48

Coordinate ProofInvolves placing geometric figures in a coordinate plane.

49

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.

50

LegsIn a right triangle, the sides

adjacent to the right angle are called the legs. (a and b)

51

HypotenuseThe side opposite the right

angle. (c)

52

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.

53

Theorem 4.12Hypotenuse-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.

54

Flow ProofUses arrows to show the flow

of a logical statement.

55

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.

56

Theorem 4.13Angle-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.

57

Chapter 5Relationships in

Triangles andQuadrilaterals

58

Midsegment of a Triangle

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

59

Theorem 5.1Midsegment 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

60

Perpendicular BisectorA segment, ray, line, or plane that is

perpendicular to a segment at its midpoint.

61

EquidistantA point is the same distance from

each of two figures.

62

Theorem 5.2Perpendicular 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.

63

Theorem 5.3Converse 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.

64

ConcurrentWhen three or more lines, rays, or

segments intersect in the same point.

65

Theorem 5.4Concurrency 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.

66

CircumcenterThe point of concurrency of the three perpendicular bisectors of a triangle.

67

Angle BisectorA ray that divides an angle into two

congruent adjacent angles.

68

IncenterPoint of concurrency of the three angle

bisectors of a triangle.

69

Theorem 5.5Angle Bisector Theorem

If a point is on the bisector of an angle, then it is equidistant from the two sides of the

angle.

70

Theorem 5.7Concurrency of Angle Bisectors of a

TriangleThe angle bisectors of a triangle intersect at a

point that is equidistant from the sides of the triangle.

71

Theorem 5.6Converse 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.

72

Median of a TriangleSegment from a vertex to the midpoint of the opposite side.

73

CentroidPoint of concurrency of the three

medians of a triangle. Always on the inside of the triangle.

74

Altitude of a TrianglePerpendicular segment from a vertex

to the opposite side or to the line that contains the opposite side.

75

OrthocenterPoint at which the lines containing the

three altitudes of a triangle intersect.

76

Theorem 5.8Concurrency 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.

77

Theorem 5.9Concurrency of Altitudes of a TriangleThe lines containing the altitudes of a

triangle are concurrent.

78

Theorem 5.10If 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.

79

Theorem 5.11If 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.

80

Theorem 5.12Triangle 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.

81

Theorem 5.13Exterior Angle Inequality Theorem

The measure of an exterior angle of a triangle is greater than the measure of either of the nonadjacent interior

angles.

82

Theorem 5.14HingeTheorem

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.

83

11cm

72

68

Theorem 5.15Converse 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.

84

11cm

7268

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 ΔABC.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 ).

85

Diagonal of a Polygon

Segment that joins two nonconsecutive vertices.

86

Theorem 5.16Polygon Interior Angles Theorem

The sum of the measures of the interior angles of a polygon is 180(n-2).

n= number of sides

87

Corollary to Theorem 5.16

Interior Angles of a QuadrilateralThe sum of the measures of the interior angles of a quadrilateral is

360°.

88

Theorem 5.17Polygon Exterior Angles TheoremThe sum of the measures of the

exterior angles of a convex polygon, one angle at each vertex, is 360°.

89

Interior Angles of the Polygon

Original angles of a polygon. In a regular polygon, the interior

angles are congruent.

90

Exterior Angles of the Polygon

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

91

ParallelogramA quadrilateral with both pairs

of opposite sides parallel.

92

Theorem 5.18If a quadrilateral is a parallelogram, then its

opposite sides are congruent.

93

Theorem 5.19If a quadrilateral is a parallelogram, then its

opposite angles are congruent.

94

Theorem 5.20If a quadrilateral is a parallelogram, then its consecutive angles are

supplementary.

95

Theorem 5.21If a quadrilateral is a parallelogram, then its

diagonals bisect each other.

96

Theorem 5.22If both pairs of opposite sides of a quadrilateral are congruent, then

the quadrilateral is a parallelogram.

97

Theorem 5.23If both pairs of opposite angles of

a quadrilateral are congruent, then the quadrilateral is a

parallelogram.

98

Theorem 5.24If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is

a parallelogram.

99

Theorem 5.25If the diagonals of a quadrilateral

bisect each other, then the quadrilateral is a parallelogram.

100

RhombusA parallelogram with four

congruent sides.

101

RectangleA parallelogram with four right

angles.

102

SquareA parallelogram with four congruent sides and four

right angles.

103

Rhombus CorollaryA quadrilateral is a Rhombus if and only if it has four congruent

sides.

104

Rectangle CorollaryA quadrilateral is a Rectangle if

and only if it has four right angles.

105

Square CorollaryA quadrilateral is a Square if and

only if it is a Rhombus and a Rectangle.

106

Theorem 5.26A parallelogram is a Rhombus if

and only if its diagonals are perpendicular.

107

Theorem 5.27A parallelogram is a Rhombus if and only if each diagonal bisects

a pair of opposite angles.

108

Theorem 5.28A parallelogram is a Rectangle if

and only if its diagonals are congruent.

109

TrapezoidA quadrilateral with exactly

one pair of parallel sides.

110

Base of a TrapezoidParallel sides of a trapezoid.

111

Legs of a TrapezoidNonparallel sides of a

trapezoid.

112

Isosceles TrapezoidTrapezoid with congruent legs.

113

Midsegment of a Trapezoid

Segment that connects the midpoints of its legs.

114

KiteA Quadrilateral that has two pairs of

consecutive congruent sides, but opposite sides are NOT congruent.

115

Theorem 5.29If a Trapezoid is Isosceles, then

each pair of base angles is congruent.

116

Theorem 5.30If a Trapezoid has a pair of

congruent base angles, then it is an Isosceles Trapezoid.

117

Theorem 5.31A Trapezoid is Isosceles if and only

if its diagonals are congruent.

118

Theorem 5.32Midsegment 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.

119

Theorem 5.33If a quadrilateral is a kite, then its

diagonals are perpendicular.

120

Theorem 5.34If a quadrilateral is a kite, then

exactly one pair of opposite angles are congruent.

121