Transcript Geometry Cliff Notes

Geometry Cliff Notes
Chapters 4 and 5
1
Chapter 4
Reasoning and Proof,
Lines, and
Congruent Triangles
2
Distance Formula
d=
( x2  x1 )  ( y2  y1 )
2
2
Example:
Find the distance between (3,8)(5,2)
(5  3) 2  (2  8) 2
( 2) 2  (  6) 2
4  36
40
d= 2 10
3
Midpoint Formula
 x1  x2 y1  y2 
,

M= 
2
 2

Example:
Find the midpoint (20,5)(30,-5)
20  30 5  5
,
2
2
M= (25,0)
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
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.
7
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
8
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.**
9
Negation
Opposite of the original statement.
Original: All sharks have a boneless skeleton.
Negation: Sharks do not have a boneless skeleton.
10
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.
11
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
12
are not an athlete. (False)
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
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basketball player. (True)
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 obtuse
18
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
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food will spoil.
Postulate
A rule that is accepted
without proof.
20
Theorem
A statement that can be
proven.
21
Subtraction Property of
Equality
Subtract a value from both
sides of an equation.
x +7 = 10
-7 -7
X=3
22
Addition Property of
Equality
Add a value to both sides of
an equation.
X-7 = 10
+7 +7
X = 17
23
Division Property of
Equality
Divide both sides by a value.
3x = 9
3 3
x=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 = 12
AB= CD
27
Proof
Logical argument that shows a
statement is true.
28
Two-column Proof
Numbered statements and corresponding
reasons that show an argument in a
logical order.
#
Statement
Reason
1
3(2x-3)+1 = 2x
Given
2
6x - 9 + 1 = 2x
Distributive Property
3
4x - 9 + 1 = 0
Subtraction Property of Equality
4
4x - 8 = 0
Add/Simplify
5
4x = 8
Addition Property of Equality
6
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
A, A  A or 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 Angles
Two 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 Angles
Congruence Theorem
Vertical angles are congruent.
1  3
or
2  4
38
Linear Pair Postulate
Two adjacent angles whose common
sides are opposite rays.
If two angles form a linear pair, then they are
supplementary.
39
Theorem 4.7
If two lines intersect to form a
linear pair of congruent angles,
then the lines are perpendicular.
B
C
A
D
ADB  CDB
40
Theorem 4.8
If two lines are perpendicular,
then they intersect to form
four right angles.
41
Theorem 4.9
If two sides of two adjacent acute
angles are perpendicular, then the
angles are complementary.
m
a
b
n
 a and  b are comple mentary
42
Transversal
A line that intersects two or
more coplanar lines at different
points.
a
b
c
c is the transve rsal
43
Theorem 4.10
Perpendicular Transversal Theorem
If a transversal is perpendicular to
one of two parallel lines, then it is
perpendicular to the other.
a
b
a
b
c
44
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.
a
b
c
45
Distance from a point
to a line
The length of the perpendicular
segment from the point to the line.
A
m
•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 Figures
All the parts of one figure are
congruent to the corresponding
parts of another figure.
(Same size, same shape)
47
Corresponding Parts
The angles, sides, and vertices that are in
the same location in congruent figures.
48
Coordinate Proof
Involves placing geometric
figures in a coordinate plane.
49
Side-Side-Side Congruence
Postulate (SSS)
If three sides of one triangle are
congruent to three sides of a second
triangle, then the two triangles are
congruent.
50
Legs
In a right triangle, the sides
adjacent to the right angle are
called the legs. (a and b)
51
Hypotenuse
The side opposite the right angle.
(c)
52
Side-Angle-Side Congruence
Postulate (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 4.12
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.
54
Flow Proof
Uses 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 4.13
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.
57
Chapter 5
Relationships in
Triangles and
Quadrilaterals
58
Midsegment of a Triangle
Segment that connects the midpoints of
two sides of the triangle.
59
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
60
Perpendicular Bisector
A segment, ray, line, or plane that is
perpendicular to a segment at its
midpoint.
61
Equidistant
A point is the same distance from
each of two figures.
62
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.

63
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.
64
Concurrent
When three or more lines, rays, or
segments intersect in the same point.
65
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.
66
Circumcenter
The point of concurrency of the three
perpendicular bisectors of a triangle.
67
Angle Bisector
A ray that divides an angle into two
congruent adjacent angles.
68
Incenter
Point of concurrency of the three angle
bisectors of a triangle.
69
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.
70
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.
71
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.
72
Median of a Triangle
Segment from a vertex to the
midpoint of the opposite side.
73
Centroid
Point of concurrency of the three
medians of a triangle. Always on the
inside of the triangle.
74
Altitude of a Triangle
Perpendicular segment from a vertex to
the opposite side or to the line that
contains the opposite side.
75
Orthocenter
Point at which the lines containing the
three altitudes of a triangle intersect.
76
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.
77
Theorem 5.9
Concurrency of Altitudes of a Triangle
The lines containing the altitudes of a
triangle are concurrent.
78
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.
79
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.
80
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.
81
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.
82
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
83
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
84
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 postulate
7) 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 5.16
Polygon Interior Angles Theorem
The sum of the measures of the interior angles
of a polygon is 180(n-2).
n= number of sides
87
Corollary to Theorem 5.16
Interior Angles of a Quadrilateral
The sum of the measures of the interior
angles of a quadrilateral is 360°.
88
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°.
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
Parallelogram
A quadrilateral with both pairs of
opposite sides parallel.
92
Theorem 5.18
If a quadrilateral is a
parallelogram, then its opposite
sides are congruent.
93
Theorem 5.19
If a quadrilateral is a
parallelogram, then its opposite
angles are congruent.
94
Theorem 5.20
If a quadrilateral is a parallelogram,
then its consecutive angles are
supplementary.
95
Theorem 5.21
If a quadrilateral is a
parallelogram, then its diagonals
bisect each other.
96
Theorem 5.22
If both pairs of opposite sides of a
quadrilateral are congruent, then
the quadrilateral is a parallelogram.
97
Theorem 5.23
If both pairs of opposite angles of a
quadrilateral are congruent, then
the quadrilateral is a parallelogram.
98
Theorem 5.24
If one pair of opposite sides of a
quadrilateral are congruent and
parallel, then the quadrilateral is a
parallelogram.
99
Theorem 5.25
If the diagonals of a quadrilateral
bisect each other, then the
quadrilateral is a parallelogram.
100
Rhombus
A parallelogram with four
congruent sides.
101
Rectangle
A parallelogram with four right
angles.
102
Square
A parallelogram with four
congruent sides and four right
angles.
103
Rhombus Corollary
A quadrilateral is a Rhombus if and
only if it has four congruent sides.
104
Rectangle Corollary
A quadrilateral is a Rectangle if and
only if it has four right angles.
105
Square Corollary
A quadrilateral is a Square if and only
if it is a Rhombus and a Rectangle.
106
Theorem 5.26
A parallelogram is a Rhombus if and
only if its diagonals are
perpendicular.
107
Theorem 5.27
A parallelogram is a Rhombus if and
only if each diagonal bisects a pair
of opposite angles.
108
Theorem 5.28
A parallelogram is a Rectangle if and
only if its diagonals are congruent.
109
Trapezoid
A quadrilateral with exactly one
pair of parallel sides.
110
Base of a Trapezoid
Parallel sides of a trapezoid.
111
Legs of a Trapezoid
Nonparallel sides of a trapezoid.
112
Isosceles Trapezoid
Trapezoid with congruent legs.
113
Midsegment of a Trapezoid
Segment that connects the midpoints
of its legs.
114
Kite
A Quadrilateral that has two pairs of
consecutive congruent sides, but
opposite sides are NOT congruent.
115
Theorem 5.29
If a Trapezoid is Isosceles, then
each pair of base angles is
congruent.
116
Theorem 5.30
If a Trapezoid has a pair of
congruent base angles, then it is an
Isosceles Trapezoid.
117
Theorem 5.31
A Trapezoid is Isosceles if and only
if its diagonals are congruent.
118
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.
119
Theorem 5.33
If a quadrilateral is a kite, then its
diagonals are perpendicular.
120
Theorem 5.34
If a quadrilateral is a kite, then
exactly one pair of opposite angles
are congruent.
121