Geometry Cliff Notes Chapters 4 and 5 1
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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
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