Download Geometry Midterm Review

Geometry Final Review
Chapter 1: Essentials of Geometry
1 - 1 Undefined Terms
Geometry - branch of math that defines and relates basic properties and measurements of line
segments and angles
Undefined Terms - meaning is accepted without definition
Set - well defined collection of items {numbers, objects, etc.}
Point - exact location in space, has no dimension
Line -
Infinite set of points in opposite directions forming straight path, 1 dimension - length
Plane - set of points forming flat surface that extends without end in all directions
1 - 2 The Real Numbers and Their Properties
Real Numbers - any number on the number line
Counting Numbers - natural numbers, 1, 2, 3, 4…
Whole Numbers - 0, 1, 2, 3…
Integers - whole numbers and opposites, …-1, 0, 1…
Rational Numbers - real number written as a fraction, can repeat and terminate 9/2, 7.12,
11.23, 3/4
Irrational Numbers - Number cannot be written as exact ratio, , 2
Rational
Integers
Whole
Counting
Irrational
Number Line - a line on which each point represents real numbers
Coordinate - ordered pair (x, y)  identifies point on coordinate plane
Graph - graphic representation used to show numeric relationships
Properties:
Closure - set of numbers is closed under operation if any two elements of the set yields another
element of set when operation (sum/product of two real numbers is a real number)
Closure property of addition - sum of two real numbers is a real number
13 + 3 = 16
1 + 79 = 80
Closure property of multiplication - product of two real numbers is a real number
3 x 8 = 24
***Subtraction 3 - 6 = -3 NOT CLOSED, thus no closure property of subtraction
Commutative:
Commutative property of addition - a + b = b + a
Commutative property of multiplication - a x b = b x a
Associative:
Associative property of addition - a + (b + c) = (a + b) + c
Associative property of multiplication - a x (b x c) = (a x b) x c
Identity Property:
Additive identity: the number 0
a+0=a
Multiplicative identity: the number 1
ax1=a
Inverse property:
Additive inverse: the opposite; gets sum to 0
a + (-a) = 0
Multiplicative inverse: 1/a; gets product to 1
a x 1/a = 1
Distributive property:
a(b + c) = ab + ac
Multiplication property of zero:
ab = 0 if and only if a = 0 or b = 0
1 - 3 Definitions, Lines, and Line Segments
Collinear set of points - set of points all of which lie on the same straight line
Non-collinear set of points - set of points not on the same straight line
Distance between two points on the real number line - absolute value of difference of coordinates of 2
points
Ex) point -3 to 5: -3 - 5 = -8 = 8
Betweenness - B is between A and C is A, B, and C are distinct collinear points,
AB + BC = AC
Line segment - set of points consisting of two points on a line, called endpoints, and all of the points in
the line between the endpoints
Length of a line segment - distance between its endpoints
Congruent segments - segments that have the same measure
1 - 4 Midpoints and Bisectors
Midpoint of a line segment - point of that line segment that divides the segment into two congruent
segments
Bisector of a line segment - any line, or subset of a line, that intersects the segment at its midpoint
Sum of two line segments - line segment RS is the sum of 2 segments, RP and PS, if P is between R and S
1 - 5 Rays and Angles
On one side of a point - A and B are on one side of the point P is A, B, and P are collinear and P is not
between A and B
Half-line - consists of the set of all points on one side of the point of division
Ray - consists of a point on a line and all points on one side of the point (one endpoint and goes
indefinitely in one direction)
Opposite Rays - 2 rays of the same line with a common endpoint and no other point in common
Angles - set of points that is the union of two rays having the same endpoint
Vertex - common endpoint of two rays that forms an angle
Side - a line segment forming adjacent vertices of a polygon
Straight angle - angle that is the union of opposite rays, 180
Acute angle - angle whose degree measure is greater than 0 and less that 90
Right Angle - angle whose degree measure is 90
Obtuse angle - angle whose degree measure is greater than 90 and less than 180
1 - 6 More Angle Definitions
Congruent Angles - angles that have the same measure
Bisector of an angle - ray whose endpoint is the vertex of the angle, and that divides that angle into two
congruent segments
Perpendicular lines - two lines that intersect to form right angles
1 - 7 Triangles
Polygon - closed figure in a plane that is union of line segments such that segments intersect only at
endpoints, no segments share common endpoint are collinear
Triangle - polygon with exactly 3 sides
Scalene Triangle - No congruent sides
Isosceles Triangle - 2 congruent sides
Legs - two congruent sides
Base - third non-congruent sides
Vertex Angle - formed by the two congruent sides
Base Angles - angles whose vertices are endpoints of base of the triangle
Equilateral Triangle - 3 congruent sides
Acute triangle - 3 acute angles
Right triangle - has a right angle
Legs - two sides of a triangle that form the right angle
Hypotenuse - side opposite the right angle
Obtuse angle - has an obtuse angle
Equiangular triangle - has 3 congruent angles
Chapter 2: Logic
2 - 1 Sentences, Statements, and Truth Values
Logic - study of reasoning
Mathematical Sentences - sentence that contains a complete thought and can be judged true or false
Ex)
Tomorrow is Tuesday
Albany is the capital of New York State
7>5
Non-Mathematical Sentences - sentence that does not state a fact, such as a command, question, or
exclamation
Ex)
Do you like math?
Find your book.
Phrase - an expression that is only part of a sentence
Ex)
All sides
3+5
Open Sentence - sentence that contains a variable
Variable - a symbol used to represent a number in
Domain or Replacement Set - set from which replacements for the variable are taken
Solution or Truth Set - set of all the replacements that will change the open sentence into a true
sentence
If no replacements from the domain make a true statement, then the solution set is { }
Closed sentence or statement - sentence that can be judged true or false
Negation: usually formed by placing the word “not” in the original statement
Symbol ~
(A statement and its negation have the opposite truth values)
P: Albany is the capital of New York State
~P: Albany is not the capital of NYS
Truth Table:
P
~P
T
F
T
F
F
T
F
T
2 - 2 Conjunctions
Conjunction - A compound statement formed by combining two simple statements using the word
“and”
Symbol ^
P: It is raining
Q: Kate is wet
P^Q: It is raining and Kate is wet
P^~Q: It is raining and Kate is not wet
~(P^~Q): It is not the case that it is raining and Kate is not wet
Truth Table:
P
T
T
F
F
Q
T
F
T
F
P^Q
T
F
F
F
Example:
P: x is odd
Q: x is prime
When x = 3
P
Q
P^Q
T
T
F
When x = 9
P
Q
P^Q
T
F
F
P
F
When x = 2
Q P^Q
T
F
When x = 6
P Q P^Q
F
F
F
2 - 3 Disjunctions
Disjunction: a compound statement formed by combining 2 simple statements using the word “or”
Symbol 
P: Pat eats pizza
Q: Carol drinks coke
PQ: Pat eats pizza or Carol drinks coke
Truth Table:
P
T
T
F
F
Q
T
F
T
F
PQ
T
T
T
F
2 - 4 Conditionals
Conditional - compound statement formed by using if  then to combine two simpler statements
Symbol 
Parts of a conditional:
Hypothesis - an assertion that begins an argument, follows if
Conclusion - ending the argument, follows then
Truth Table:
P
T
T
F
F
Q
T
F
T
F
PQ
T
F
T
T
Hidden Conditional: If, then aren’t in the statement, but suggests conditional
Ex) When I finish my homework, I’ll go to the movies.
2 - 5 Inverses, Converses, and Contrapositives
Inverse - Formed by negating the hypothesis and conclusion
Conditional: If a number is not a whole number, then it is not an integer
Inverse: If a number is a whole number, then is it an integer
Truth Table:
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
PQ
T
F
T
T
~P~Q
T
T
F
T
Converse - formed by interchanging the hypothesis and the conclusion
Conditional: If a number is an integer, then it is a whole number
Converse: If a number is a whole number, then it is an integer
Truth Table:
P
T
T
F
F
Q
T
F
T
F
PQ
T
F
T
T
QP
T
T
F
T
Contrapositive - formed by negating and interchanging both the hypothesis and the conclusion
Conditional: If a number is not an integer, then it is not a whole number
Contrapositive: If a number is a whole number, then a number is an integer
Truth Table:
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
PQ
T
F
T
T
~Q~P
T
F
T
T
Logically Equivalents - having the same truth value
P
T
T
F
F
Q
T
F
T
F
~P
F
F
T
T
~Q
F
T
F
T
PQ
T
F
T
T
~P~Q
T
T
F
T
QP
T
T
F
T
~Q~P
T
F
T
T
Conditional (PQ) is logically equivalent to Contrapositive (~Q~P)
Inverse (~P~Q) is logically equivalent to Converse (QP)
2 - 6 Biconditionals
Biconditional - conjunction of the conditional and its converse
Symbol: PQ^QP, PQ
Example:
P: A polygon is a triangle
Q: A polygon has exactly 3 sides
PQ: A polygon is a triangle is and only if it has 3 sides
Truth Table:
P
Q
T
T
T
F
F
T
F
F
PQ
T
F
T
T
QP PQ^QP
T
T
T
F
F
F
T
T
PQ
T
F
F
T
PQ is true when: P and Q are both true or both false
2 - 7 Laws of Logic
The Law of Detachment - If a conditional is true and the hypothesis is true then the conclusion is true
Valid Argument - uses a series of statements called premises that have known truth values to arrive at
the conclusion
Ex)
“If today is Friday, then I get my allowance” T
“Today is Friday” T
Thus: I get my allowance is True
The Law of Disjunctive Inference - If the disjunction (pq) is true and the disjunct (p) is false, then the
other disjunct (q) is true
2 - 8 Drawing Conclusions
Example:
1. If Rachel joins the choir, then Rachel likes to sing
T
2. Rachel will join the choir or Rachel will play basketball
T
3. Rachel does not like to sing
T
Conclusions:
Rachel will play basketball (She won’t join choir and doesn’t like to sing)
Example:
Ted, Bill, and Mary each take a different course in one of three areas for their senior year:
mathematics, art, and thermodynamics.
1. Ted tutors his sibling taking the mathematics course
2. The art students and Ted have an argument over last night’s basketball game
3. Mary loves the drawing made by her sibling taking the art course
T
B
M
M
X
X

A
X

X
T

X
X
Tautology - logical statement that is always true for all truth values of the simple statements that
compose it
Chapter 3: Proving Statements in Geometry
3 - 1 Inductive Reasoning
Generalization - Going from a few specific cases to a general case
Inductive Reasoning - method of reasoning in which a series of particular examples leads to a
conclusion
Counterexample - Example that disproves a generalization
Conjectures - statements that are likely to true but have not yet been proven true by a deductive
proof
3 - 2 Definitions as Biconditionals
Deductive Reasoning - uses the laws of logic to combine definitions and general statements that we
know to be true to reach a valid conclusion
- A scalene triangle is a triangle that has no congruent sides
- If a triangle is scalene, then the triangle has no congruent sides
- A triangle is scalene if and only if the triangle has no congruent sides
3 - 3 Deductive Reasoning
Proofs - In geometry, it is a valid argument that establishes the truth of a statement
Perpendicular Lines - 2 lines that intersect to form right angles
Parallel Lines - Coplanar lines that have no points or all points in common
Line Segment - A set of points consisting of endpoints and all the points in between
Angle Bisector - A line extending from the vertex of an angle, and it divides the segment into two
congruent segments
Ray - A line extending from one endpoint indefinitely
Congruent angles - Two angles with the same measure
Congruent segments - segments with the same measure
Proof Format in Euclidean Geometry:
Given:
Prove:
Construction of a two-column proof
Examples:
1)
Given: In ABC, AB is perpendicular to BC
Prove: ABC is a right angle
Statements
Reasons
1. AB is perpendicular to BC
1. Given
2. ABC is a right angle
2. If 2 lines are perpendicular, they form right angles
2)
Given: BD is the bisector of ABC
Prove: mABD = mDBC
Statements
1. BD is the bisector of ABC
2. mABD = mDBC
Reasons
1. Given
2. If an angle is bisected then it is divided
into two congruent segments
3 - 4 Direct and Indirect Proofs
Direct Proofs - proof that starts with a given statement and uses laws of logic to arrive at the statement
to be proved
Example:
Given: ABC is an acute triangle
Prove: mA < 90
Statements
1. ABC is an acute triangle
2. A, B, C are acute angles
3. A < 90
Reasons
1. Given
2. An acute triangle has 3 acute angles
3. If an angle is acute, its measure is
greater than 0 and less than 90
Indirect Proofs - Proof that starts with the negation of a statement to be proved false
Example:
Given: AB and CD such that AB  CD
Prove: AB and CD are not congruent segments
Statements
Reasons
1. AB and CD are congruent
1. Assumption
segments
2. AB = CD
2. Congruent segments have the same
measure
3. Given
4. Contradiction in 2 and 3
Therefore, the assumption is false and the
negation is true
3. AB  CD
4. AB and CD are not congruent
segments
3 - 5 Postulates, Theorems, and Proof
Postulate (axiom) - statement whose truth is accepted without proof
Theorems - statement that is proved by deductive reasoning
Reflexive Postulate - a quantity is equal to itself
Ex) 7 = 7, x = x
Symmetric Postulate - equality may be expressed in either order
Ex) 7 = x, x = 7
Transitive Postulate - quantities equal to the same quantity are equal to each other
Ex) x = y, y = z, x = z
Example:
Given: AB = CD
A
BC = AD
AB = BC
Prove: AD = CD
D
Statements
1. AB = CD, BC = AD, AB = BC
2. CD = BC
3. CD = AD
B
C
Reasons
1. Given
2. Quantities equal to the same quantity
are equal to each other
3. Transitive; same as 2
3 - 6 The Substitution Postulate
Substitution Postulate - a quantity may be substituted for its equal in any statement of equality
Example:
Given: ABD + DBC = 90
ABD = CBE
Prove: CBE + DBC = 90
Statements
1. ABD + DBC = 90
ABD = CBE
2. CBE + DBC = 90
A
B
D
C
E
Reasons
1. Given
2. A quantity may be substituted for its
equal in any statement of equality
3 - 7 The Addition and Subtraction Postulates
Partition Postulate - a whole is equal to the sum of its parts
Addition Postulate - If equal quantities are added to equal quantities, then the sums are equal
A
B
C
Example:
Given: ABC and DEF with AB = DE and BC = EF D
E
F
Prove: AC = DF
Statements
1. AB = DE and BC = EF
2. AB + BC = DE + EF
3. AC = DF
Reasons
1. Given
2. If equal quantities are added to equal
quantities, then the sums are equal
3. A quantity may be substituted for its
equal in any statement of equality
Subtraction Postulate - If equal quantities are subtracted from equal quantities, the differences are
equal
Example:
Given: x + 6 = 14
Prove: x = 8
Statements
1. x + 6 = 14
2. x + 6 - 6 = 14 - 6
3. x = 8
Reasons
1. Given
2. If equal quantities are subtracted from
equal quantities, then the differences are
equal
3. A quantity may be substituted for its
equal in any statement of equality
3 - 8 The Multiplication and Division Postulates
Multiplication Postulate - If equal quantities are multiplied by equal quantities, then the products are
equal
Doubles of equal quantities are equal
Division Postulate - If equal quantities are divided by equal quantities, then the quotients are equal
Halves of equal quantities are equal
Powers Postulate - The squares of equal quantities are equal
Roots Postulate - If a = b and a > 0, then a = b
Positive square roots of positive equal quantities are equal
Examples:
1.
Given: AB = CD, RS = 3AB, LM = 3CD
Prove: RS = LM
Statements
1. AB = CD, RS = 3AB, LM = 3CD
2. 3AB = 3CD
3. RS = LM
2.
Given: 5x + 3 = 38
Prove: x = 7
Statements
1. 5x + 3 = 38
2. 5x + 3 - 3 = 38 - 3
3. 5x = 35
Reasons
1. Given
2. If equal quantities are multiplied by equal
quantities, then the
products are equal
3. A quantity may be substituted for
its equal in any statement of equality
Reasons
1. Given
2. If equal quantities are subtracted
from equal quantities, then the
differences are equal
3. A quantity may be substituted for
its equal in any statement of equality
4. 5x  5 = 35  5
4. If equal quantities are divided by
equal quantities, then the quotients
are equal
5. A quantity may be substituted for i
its equal in any statement of equality
5. x = 7
Chapter 4: Congruence of Line Segments, Angles, and Triangles
4 - 1 Postulates of Lines, Line Segments, and Angles
A line segment can be extended to any length in either direction
Through two given points, one and only one line can be drawn
Two lines cannot intersect in more than one point
One and only one circle can be drawn with any given point as center and the length of any given
line segment as a radius
At a given point on a given line, one and only one perpendicular can be drawn to the line
From a given point not on a given line, one and only one perpendicular can be drawn to the line
For any two distinct points, there is only one positive real number that is the length of the line segment
joining the two points
The shortest distance between two points is the length of the line segment joining the two points
A line segment has one and only one midpoint
An angle has one and only one bisector
Conditional Statements: The information that is known to be true is often stated as given and what is
to be proved as prove. When the information needed for a proof is presented in a conditional
statement, we use the information in the hypothesis to form a given statement, and the information
in the conclusion to form a prove statement.
Examples:
1. If a ray bisects a straight angle, it is perpendicular to the line determined by the straight
angle
Given: BD bisects ABC
ABC is a straight angle
Prove: BD is perpendicular to AC
2. If AB = AD and DC = AD, then AB = DC
Given: AB = AD, DC = AD
Prove: AB = DC
Then write a formal proof…
4 - 2 Using Postulates and Definitions in Proofs
Given: AB = CD
Prove: AC = BD
Statements
1. AB = CD
2. BC = BC
3. AB + BC = BC + CD
4. AC = BD
A
B
C
D
Reasons
1. Given
2. A quantity is equal to itself
3. If equal quantities are added to equal
quantities, then the sums are equal
4. A quantity may be substituted for its
equal in any statement of equality
A
E
Given: AE = AF, EB = FC
Prove: ABC is isosceles
Statements
1. AE = AF, EB = FC
2. AE + EB = AF + FC
B
3. AB = AC
4. ABC is isosceles
F
C
Reasons
1. Given
2. If equal quantities are added to equal
quantities, then the sums are equal
3. A quantity many be substituted for its
equal in any statement of equality
4. An isosceles triangle has 2 congruent
sides
4 - 3 Proving Theorems About Angles
If two angles are right angles, then they are congruent
If two angles are straight angles, then they are congruent
Adjacent Angles - two angles in the same plane that have a common vertex and a common side
but do not have any interior points in common
Complementary Angles - two angles, the sum of whose degree measures is 90
Supplementary Angles - two angles, the sum of whose degree measures is 180
Example:
Given: 1 = 2
Prove: 3 = 4
Statements
1. 1 = 2, ABCD
2. ABC and DCB are straight
angles
3. ABC = DCB
4. ABC - 1 = DCB - 2
5. 3 = 4
A
3 1
2 4
B
C
D
Reasons
1. Given
2. If ABCD, then ABC and DCB are straight
angles
3. All straight angles are congruent
4. If equal quantities are subtracted from equal
quantities, then the differences are equal
5. A quantity may be substituted for its equal
in any statement of equality
If two angles are complements of the same angle, then they are congruent
If two angles are congruent, then their complements are congruent
If two angles are supplements of the same angle, then they are congruent
If two angles are congruent, then their supplements are congruent
Linear Pair of Angles - two adjacent angles whose sum is a straight angle
If two angles form a linear pair, then they are supplementary
If two lines intersect to form congruent adjacent angles, then they are perpendicular
Vertical Angles - the non-adjacent angles formed by the intersection of two lines
If two lines intersect, then the vertical angles are congruent
A
D
Example:
Given: ABC and DBE intersect at B
B
BC bisects EBF
F
Prove: CBF = ABD
E
C
Statements
Reasons
1. ABC and DBE intersect at B
1. given
BC bisects EBF
2. EBC = CBF
2. If an angle is bisected, two congruent
angles are formed
3. EBC and ABD are vertical angles
3. Vertical angles are the two nonadjacent angles formed by the
intersection of two lines
4. EBC = ABD
4. If two lines intersect, then the vertical
angles are congruent
5. CBF = ABD
5. Quantities equal to the same quantities
are equal
Example:
WX and YZ intersect at V, WVY = 3x + 25 and XVZ = 10x + 4. Find WVZ
W
Y
Let WVY = 3x + 25
Let XVZ = 10 x + 4
Let WVZ = 180 - 10x + 4
Z
V
X
3x + 25 = 10x + 4
21 = 7x
3=x
The measure of angle WVZ is 146
10 (3) + 4 = 34
180 - 34 = 146
4 - 4 Congruent Polygons and Corresponding Parts
Congruent Polygons - polygons that have the same size and shape
One-to-one correspondence - each part or a polygon corresponds to exactly one part of the
congruent polygon
Any geometric figure is congruent to itself
A congruence may be expressed in either order
Two geometric figures congruent to the same geometric figure are congruent to each other
4 - 5 Proving Triangles Congruent Using Side, Angle, Side
SAS - Two triangles are congruent if two sides and the included angle of one triangle are congruent,
respectively, to two sides and the included angle of the other
A D
Given: BD bisects ABC, AB = BC
Prove: ADB = CDB
Statements
1. BD bisects ABC, AB = BC
2. ABD = CBD
3. BD = BD
4. ADB = CDB
C
Reasons B
1. Given
2. If an angle is bisected, it is divided into
two congruent angles
3. A quantity is equal to itself
4. SAS = SAS
4 - 6 Proving Triangles Congruent Using Angle, Side, Angle
ASA - Two triangles are congruent if two angles and the included side of one triangle are congruent,
respectively, to two angles and the included side of the other
Given: AB is perpendicular to BC
ED is perpendicular to DC
C is the midpoint of BD
Prove: ABC = EDC
Statements
1. AB is perpendicular to BC
ED is perpendicular to DC
C is the midpoint of BD
2. B and D are right angles
3. B = D
4. BC = CD
5. ACB and ECD are vertical s
6. ACB = ECD
7. ABC = EDC
A
C
D
B
E
Reasons
1. Given
2. Perpendicular lines form right angles
3. All right angles are congruent
4. A midpoint divides a segment into 2
congruent segments
5. Vertical angles are two non-adjacent
angles formed by the intersection of 2 lines
6. All vertical angles are congruent
7. ASA = ASA
4 - 7 Proving Triangles Congruent Using Side, Side, Side
SSS - Two triangles are congruent is the three sides of one triangle are congruent, respectively, to the
three sides of the other
K
Given: Isosceles triangle JKL
JK = KL
M is the midpoint of JL
J
L
Prove: JKM = LKM
M
Statements
Reasons
1. JK = KL
1. Given
M is the midpoint of JL
2. JM = ML
2. A midpoint divides a segment into two
congruent segments
3. KM = KM
3. A quantity is equal to itself
4. JKM = LKM
4. SSS = SSS
Chapter 5: Congruence Based on Triangles
5 - 1 Line Segments Associated with Triangles
Altitude of a Triangle - line segment drawn from any vertex of a triangle perpendicular to and ending in
the line that contains the opposite side
Median of a Triangle - line segment that joins any vertex of a triangle to the midpoint of the opposite
side
Angle bisector of a Triangle - ray whose endpoint is vertex of the angle and divides the angle into two
congruent segments
Examples:
B
Given: BD is the median of AE
InBDC, BE is the median to DC
A = C
AB = BC
A
C
Prove: ABD = CBE
D
E
Statements
Reasons
1. BD is the median of AE
1. Given
BE is the median to DC
A = C
AB = BC
2. D is the midpoint of AE
2. A midpoint is a line segment that joins
E is the midpoint of DC
vertex of a triangle to the midpoint of the
Opposite side
3. AD = DE, DE = EC
3. A midpoint divides a segment into two
congruent segments
4. AD = EC
4. Quantities equal to the same quantity
are equal to each other
5. ABD = CBE
5. SAS = SAS
B
Given: BD is the altitude of ABC
D is the midpoint of AC
Prove: ABD = CBD
Statements
1. BD is the altitude of ABC
D is the midpoint of AC
2. AD = DC
3. BD is perpendicular to AC
4. ADB and CDB are right angles
5. ADB = CDB
6. BD = BD
7. ABD = CBD
Reasons
1. Given
A
D
C
2. A midpoint divides a segment into
two congruent segments
3. An altitude of a triangle extends from
the vertex of the triangle perpendicular
to and ending in the opposite side
4. Perpendicular lines form right angles
5. All right angles are congruent
6. A quantity is equal to itself
7. SAS = SAS
5 - 2 Using Congruent Triangles to Prove Line Segments Congruent and Angles Congruent
If two triangles are congruent, then their corresponding parts are congruent
C
Given: ABC, CA = CB, AD = BD
Prove: ACD = BCD
Statements
1. CA = CB, AD = BD
2. CD = CD
3. CAD = CBD
4. ACD = BCD
5 - 3 Isosceles and Equilateral Triangles
A
B
Reasons
D
1. Given
2. A quantity is equal to itself
3. SSS = SSS
4. Corresponding parts of congruent are
congruent
If two sides of a triangle are congruent, the angles opposite these sides are congruent
The median from the vertex angle of an isosceles triangle bisects the vertex angle
The median from the vertex angle of an isosceles triangle is perpendicular to the base
Every equilateral triangle is equiangular
Given:  ABC is isosceles with vertex angle B
B
AD = CD
Prove: BAD = BCD
A
C
Statements
Reasons
D
1. ABC is isosceles with vertex angle B 1. Given
AD = CD
2. BA = BC
2. An isosceles triangle has 2 congruent
legs surrounding the vertex angle
3. BAC = BCA, ACD = CAD
3. Angles opposite congruent sides are
congruent
4. BAC + CAD = BCA + ACD
4. If equal quantities are added to equal
quantities, then the sums are equal
5. BAD = BCD
5. A quantity may be substituted for its
equal in any statement of equality
5 - 4 Using Two Pairs of Congruent Triangles
C
Given: AEB, AC = AD
CB = DB E
Prove: CE = DE
Statements
1. AC = AD, CB = DB
2. AB = AB
3. ACB = ADB
4. CBE = DBE
5. EB = EB
6. CEB = DEB
7. CE = DE
A
B
D
Reasons
1. Given
2. A quantity is congruent to itself
3. SSS = SSS
4. CPCTC
5. A quantity is equal to itself
6. SAS = SAS
7. CPCTC
5 - 5 Proving Overlapping Triangles Congruent
G
Given: GA = GC, AR = CE
Prove: GAE = GCR
R
A
Statements
1. GA = GC, AR = CE
2. GA - AR = GC - CE
3. RG = EG
4. G = G
5. GAE = GCR
E
C
Reasons
1. Given
2. If equal quantities are subtracted from
equal quantities, then the differences are
equal
3. Substitution postulate
4. A quantity is equal to itself
5. SAS = SAS
5 - 6 Perpendicular Bisector of a Line Segment
Perpendicular Bisector of a Line Segment - any line that is perpendicular to the line segment at its
midpoint
Concurrent - lines that intersect in one point
Intersection of the perpendicular bisector of the three sides
Perpendicular Bisector Concurrence Theorem - perpendicular bisectors of a triangle are concurrent
Circumcenter - point where the 3 perpendicular bisectors of the sides of the triangle intersect
P
Given: PQ is the perpendicular bisector of RS
Prove - RSP is isosceles R
Statements
1. PQ is the perpendicular bisector of RS
2. PQR and PQS
are right angles
Q
S
Reasons
1. Given
2. Perpendicular lines form right
angles
3. All right angles are congruent
4. A quantity is equal to itself
5. A perpendicular bisector extends
to the midpoint of the opposite side
6. A midpoint divides a segment into
two congruent segments
7. SAS = SAS
8. CPCTC
9. An isosceles triangle has 2
congruent sides
3. PQR = PQS
4. PQ = PQ
5. Q is the midpoint of RS
6. RQ = QS
7. PQR = PQS
8. PR = PS
9. RPS is isosceles
5 - 7 Basic Constructions
Construction 1: Construct a Line Segment Congruent to a Given Line Segment
Steps:
1. With a straight edge draw a ray
2. Open the compass so that the point is on A and the point of the pencil is on B
3. Using the say compass radius, draw an arc that intersects the ray
A
B
Construction 2: Construct an Angle Congruent to a Given Angle
Steps:
1. Draw a ray with a straight edge with the endpoint D
2. With A as the center, draw an arc that intersects each ray of angle A, using the same radius,
draw an arc with D as the center. Where the arc and the ray intersect, call that E
3. With E as the center, draw an arc, draw an arc with the radius equal to BC that intersects that
arc in step 3, Label it F
4. Draw DF
C
A
B
Construction 3: Construct the Perpendicular Bisector of a Given Line Segment and the Midpoint of A
given Line Segment
Steps:
1. Open the compass to a radius greater than one-half of AB
2. Place the point of the compass at A and draw an arc above AB and an arc below AB
3. Using the dame radius, place the point of the compass at B and draw an arc
above AB and an arc below AB
4. Using a straight edge, draw CD intersecting AB
A
B
Construction 4: Bisect a Given Angle
Steps:
1. With B as center, draw an arc that intersects ray BA at D and ray BC at E
2. With D as the center, draw an arc in the interior or angle ABC
3. Using the same radius, and with E as the center, draw an arc that intersects the arc drawn in
step two. Label it F.
4. Draw ray BF
A
B
C
Construction 5: Construct a Line Perpendicular to a Given Line Through a Given Point on a Line
Steps:
1. With P as the center draw arcs that intersect PA at C and PB at D
2. With C and D as centers and a radius greater than that used in step 1, draw arcs intersecting
at E
3. Draw EP
A
P
B
Construction 6: Construct a Line Perpendicular to a Given Line Through a Point Not on a Given Line
Steps:
1. With P as the center, draw an arc that intersects AB in two points, C and D
2. Open the compass to a radius greater than one-half of CD. With C and D as centers, draw
intersecting arcs. Label the point of intersection E.
3. Draw line PE intersecting line AB
A
B
Chapter 6: Transformations and the Coordinate Plane
6 -1 The Coordinates of a Point in a Plane
x - axis - horizontal line
y - axis - vertical line
Origin - point where x and y meet
Coordinate - every point on the plane can be described by 2 numbers
Ordered Pair - (x,y)
Abscissa - x coordinate
Ordinate - y coordinate
Two points are on the same horizontal line if and only if they have the same y coordinates
The length of the horizontal line segment is the absolute value of the difference of the x coordinates
Two points are on the same vertical line if and only if they have the same x coordinates
The length of a vertical line segment is the absolute value of the difference of the y coordinates
Each vertical line is perpendicular to each horizontal line
To find area of graphed objects: find internal shapes
6 - 2 Line Reflections
Line of Reflection - line in a picture that when folded makes the object coinside
Life Reflection - one to one correspondence between an object and the image points
Fixed Points - point on the line of reflection
Transformation - one to one correspondence between the set of points in S and S’ such that every point
in S corresponds to one point one point in S’
Image - S’ is called the image of the points in S
Preimage - S in called the preimage of S’
***Under a line reflection, distance, angle measure, collinearity, and midpoint are preserved
Line Symmetry - figure is its own image under a line reflection
Examples:
WOW
KICK
unlimited
both ways
6 - 3 Line reflections in the Coordinate Plane
Rules:
Line reflection in the y-axis: (x, y)  (-x, y)
Line reflection in the x-axis: (x, y)  (x, -y)
Line reflection in y = x: (x, y)  (y, x)
6 - 4 Point Reflections in the Coordinate Pane
***Under point reflection, distance, angle measure, collinearity, and midpoint is preserved
Point Symmetry - a figure has point symmetry if the figure is its own image under a reflection in a point
Reflection of the origin:
Ro (a, b)  (-a, -b)
6 - 5 Translations in the Coordinate Plane
Translation - transformation of the plane that moves every point in the plane the same distance in the
same direction
***Under a translation, distance, angle measure, collinearity, and midpoint is preserved
Translational Symmetry - if the image of every point of a figure is a point in the figure
Rule:
Ta,b (x, y)  (x + a, y + b)
6 - 6 Rotations in the Coordinate Plane
Rotations - transformation of the plane about a fixed point P through an angle of d degrees such that a
counter clockwise rotation is positive and a clockwise rotation is negative
Rotation Symmetry - If a figure is its own image under a rotation and the center of the rotation is the only
fixed point
Rules:
R90 (x, y)  (-y, x)
R180 (x, y)  (-x, -y)
R270 (x, y)  (y, -x)
Thus…R360 (x, y)  (x, y)
6 - 7 Glide Reflections
Glide Reflection - composition of transformations of the plane that consist of a line reflection and a
translation in the direction of the line of reflection performed in either order
Isometry - transformation that preserves distance
6 - 8 Dilations in the Coordinate Plane
Dilations - a transformation in the plane that preserves angle measure but not distance
Rule:
Dk (x, y)  (k x x, k x y)
6 - 9 Transformations as Functions
Function - set of ordered pairs in which no 2 pairs have the same first element
Domain - in an ordered pair, the set of first elements is the domain of the function
Range - In an ordered pair, the set of second elements is the range of the function
Composition of Transformations - combination of 2 transformations in which the first transformation
produces an image and the second performs on that image
Orientation - order of points around an object
Direct Isometry - transformation that preserves distance and orientation
Opposite Isometry - transformation that preserves distance but changes the order of orientation from
counterclockwise to clockwise or visa versa ex) line reflection
Chapter 7: Geometric Inequalities
7 - 1 Basic Inequality postulates
Whole Quantity and Its Parts - The whole is greater than any of its parts
Transitive Property of Inequalities - a, b, and c are real numbers such that a>b and b>c then a>c
Substitution Postulate of Inequalities - A quantity my be substituted for its equal in any statement of
inequality
Trichotomy Postulate - a and b are real numbers then one of the following must be true: a<b, a = b, or
a>b
7 - 2 Inequality Postulates Involving Addition and Subtraction
Addition Postulate of Inequalities:
A: I equal quantities are added to unequal quantities, the sums are unequal in the same order
B: If unequal quantities are added to unequal quantities of the same order, the sums are
unequal in the same order
Subtraction postulate of Inequalities - If equal quantities are subtracted from unequal quantities, the
differences are unequal in the same order
7 - 3 Inequality Postulates Involving Multiplication and Division
Multiplication Postulates of Inequalities A: If unequal quantities are multiplied by positive equal quantities, then the products are
unequal in the same order
B: If unequal quantities are multiplied by negative equal quantities, then the products are
unequal in the opposite order
Division Postulate of Inequalities A: If unequal quantities are divided by positive equal quantities, then the quotients are
unequal in the same order
B: If unequal quantities are divided by negative equal quantities, then the quotients is
unequal in the opposite order
7 - 4 An inequality Involving the Lengths of the Sides of a triangle
Triangle Inequality Theorem - The length of one side of the triangle is less than the sum of the lengths of
the other 2 sides
7 - 5 An Inequality Involving an Exterior Angle of a Triangle
Exterior Angle of a Polygon - angle that forms a linear pair with one of the interior angles of the polygon
Adjacent Interior angle - for each interior angle there is an adjacent interior angle
Remote (non-adjacent interior angles) - other two angles in the triangle
The measure of an exterior angle of a triangle is greater than the measure of either non-adjacent
interior angles
7 - 6 Inequalities Involving Sides and Angles of a Triangle
If the lengths of two sides of a triangle are unequal, then the measures of the angles opposite these
sides are unequal and the larger angle lies opposite the longer side
If the measure of two angles of a triangle are unequal, then the lengths of the sides opposite these
angles are unequal and the longer side lies opposite the larger angle
Chapter 8: Slopes and Equations of Lines
8 - 1 The Slope of A Line
Finding the Slope of a Line:
M = y = rise = y2 - y1
x
run x2 - x1
To find, select two points on a line. Find the vertical change, then the horizontal. Write the
ratio of change.
Positive Slope
Negative Slope
Zero Slope
No/Undefined Slope
8 - 2 The Equation of a Line
Y-intercept - y-coordinate of the point at which the line intersects the x-axis
X-intercept - x-coordinate of the point at which the line intersects the y-axis
A, B, and C lie on the same line if and only if the slope of AB is equal to the slope of BC
To write the equation of a line:
First, find the slope of the line. Then, plug this slope (m) and one coordinate (x, y) into the y=
m x + b formula. Solve for b. Your equation will be something like y = 2x + 1
Point-Slope Formula:
Y-b=m
X-a
8 - 3 Midpoint of a Line Segment
Midpoint - point of a line segment that divides the segment into 2 congruent segments
M=
X 1 + X 2, Y 1 + Y 2
2
2
8 - 4 The Slope of Perpendicular Lines
If two non-vertical lines are perpendicular, then the slope of one is the negative reciprocal of the other
8 - 5 Coordinate Proof
Example 1: Prove that AB and CD bisect
each other and are perpendicular to
each other if the coordinates of the
endpoints of these segments are
A (-3,5), B (5,1), C (-2,-3), D (4,9).
The Midpoint of AB and CD is (1,3).
The slope of AB is -1/2 and of CD
Is 2. Since these are negative reciprocals, And
the two lines intersect at their midpoint, AB and
are
Perpendicular and bisect each other.
QuickTi me™ and a
TIFF ( Uncompressed) decompr essor
are needed to see thi s p icture.
since
BC
8 - 6 Concurrence of the Altitudes of a Triangle
Concurrent - 3 or more lines are concurrent is they
intersect in one point
The altitudes of a triangle are concurrent
Orthocenter - point at which altitudes of a triangle intersect
To find, find the negative reciprocal slope for each side of the triangle. Use this slope and plug it
into the point that is opposite the side for which it applies. Plug into y = m x + b to find equation of
line. You can plug these into a system algebraically (if two things equal y, set them equal to each
other) or simply graph.
This will give you the point of orthocenter
Example 1) The Coordinates of the
Vertices of PQR are P (0,0), Q (-2,6),
And R (4,0). Find the orthocenter.
mPQ = -3
m(alt) = 1/3
0 = 1/3 (4) + b
0 = 4/3 + b
- 4/3 = b
y = 1/3x + -4/3
mQR = -1
m (alt) = 1
0 = 1 (0) + b
b=0
y=x
mPR = 0
m (alt) = undefined
-2 = x
y=x
y = -2
x = -2
QuickTi me™ and a
TIFF ( Uncompressed) decompr essor
are needed to see thi s p icture.
Orthocenter = (-2,-2)
Chapter 9: Parallel Lines
9 - 1 Proving Lines Parallel
Coplanar - all points/lines are in the same plane
Parallel Lines - lines in the same plane and that have no points in common or have all points in
common (coincide) AB and CD
Transversal - intersects two other coplanar lines,
EF
Interior angles - in between two given lines,
3, 4, 5, 6
QuickTi me™ and a
TIFF ( Uncompressed) decompressor
are needed to see thi s pi ctur e.
Exterior angles - on outside of given lines,
1, 2, 7, 8
Alternate interior angles - on opposite sides of transversal and
common vertex,
3 & 6, 4 & 5
don’t have
Alternate exterior angles - same but on exterior, 1 & 8, 2 & 7
Interior angles on the same side of the transversal - equal 180, 3 & 5, 4 & 6
Corresponding angles - same side of transversal and don’t have common vertex, one exterior and
one interior, 1 & 5, 3 & 7, 2 & 6, 4 & 8
9 - 2 Properties of Parallel Lines
If two coplanar lines are cut by a transversal so that the alternate interior angles formed are
congruent, then the two lines are parallel
If two coplanar lines are cut by a transversal so that the corresponding angles are congruent, then
the two lines are parallel
If two coplanar lines are cut by a transversal so that the interior angles on the same side of the
transversal are supplementary, then the lines are parallel
If a transversal is perpendicular to one of two parallel lines, it is perpendicular to the other
If two of three lines in the same plane are parallel to the third line, then they are parallel to each
other
9 - 3 Parallel Lines in the Coordinate Plane
If two non-vertical lines in the same plane are parallel, then they have the same slope
9 - 4 The Sum of the Measures of the Angles of the Triangle
The sum of the measure of the angles of a triangle is 180
If two angles of one triangle are congruent to 2 angles of another triangle, the third angles are
congruent
The acute angles of a right triangle are complementary
Each angle of an equilateral triangle measures 60
Each acute angle of an isosceles right triangle measures 45
The sum of the measure of the angles of a quadrilateral is 360
The measure of an exterior angle of a triangle is equal to the sum of the measures of the
nonadjacent interior angles
9 - 5 Proving Triangles Congruent by Angle, Angle, Side
If two angles and the sides opposite one of them in one triangle are congruent to the
corresponding angles and sides in another triangle, then the triangles are congruent
***Two triangles cannot be proven congruent using AAA or SSA
9 - 6 The Converse of the Isosceles Triangle Theorem
If two angles of a triangle are congruent, then the sides opposite these angles are congruent
9 - 7 Proving Right Triangles Congruent by Hypotenuse Leg
If the hypotenuse and a leg of one triangle are congruent to the corresponding parts of the other,
then the two right angles are congruent
The angle bisectors of a triangle are concurrent
Incenter - the point where the angle bisectors meet
9 - 8 Interior and Exterior Angles of Polygons
Polygon - closed figure
Triangle - 3 sides
Quadrilateral - 4 sides
Pentagon - 5 sides
Hexagon - 6 sides
Octagon - 8 sides
Decagon - 10 sides
n-gon - variable
Convex Polygon - polygon in which at least one angle measures under 180
Concave Polygon - polygon in which at least one angle measures over 180
Diagonal - a lone segment whose endpoints are 2 non-adjacent vertices
The sum of the degree measures of the interior angles of any polygon of n sides is 180(n-2)
The sum of the measure of the exterior angles of a polygon is 360
Regular Polygon - polygon that’s equilateral and equilngular
Chapter 10: Quadrilaterals
10 - 2 The General Quadrilateral / The Parallelogram
Quadrilateral - polygon with 4 sides
Q
R
P
S
Consecutive Vertices - (Q,P) (Q,R) (R,S) (S,P)
Consecutive Sides - PS, SR, PQ, QR
Parallelogram - quadrilateral in which two pairs of opposite sides are parallel and congruent
10 - 3 Proving a Quadrilateral is a Parallelogram
-
Both pairs of opposite sides are parallel
Both pairs of opposite sides are congruent
Two opposite sides are congruent and parallel
The diagonals bisect each other
Both pairs of opposite angles are congruent
10 - 4 The Rectangle
Rectangle - parallelogram with one right angle
A quadrilateral is a rectangle if any of the following can be proven true:
-
It is a parallelogram with one right angle
It is equiangular
It is a parallelogram whose diagonals are congruent
10 - 5 Rhombus
Rhombus - parallelogram with 2 congruent consecutive sides
A quadrilateral is a rhombus if any of the following can be proven true:
-
It is a parallelogram with two congruent consecutive sides
It is equilateral
It is a parallelogram whose diagonals are perpendicular to each other
It is a parallelogram whose diagonals bisect opposite angles
10 - 6 Square
A quadrilateral is a square if any of the following can be proven true:
-
It is a rectangle with two consecutive sides congruent
It is a rhombus with one right angle
10 - 7 Trapezoid
Trapezoid - quadrilateral in which two and only two sides are parallel
Isosceles Trapezoid - quadrilateral in which the non-parallel sides are congruent
Median of a Trapezoid - line segment whose endpoints are the midpoints of the non-parallel sides of a
trapezoid
A trapezoid is an isosceles trapezoid is any of the following can be proven true:
-
Base angles of a trapezoid are congruent
Diagonals of a trapezoid are congruent
Consecutive non-base angles are supplementary
The median is parallel to the bases
10 - 8 Area of Polygons
Area of a Polygon - unique real number assigned to any polygon that indicates the number of the nonoverlapping square units contained in polygon interior
Formulas:
Rectangle: a = L X W
Triangle: a = ½ b X h
Square: A = S2
Trapezoid: a = 1/3 (b1 + b2) h
Parallelogram: a = b X h
Rhombus: a = b X h; a = ½ X d1 X d2
Chapter 11: The Geometry of Three Dimensions
11 - 1 Points, Lines, and Planes, Perpendicular Lines and Planes, Parallel Lines and Planes
Parallel Lines in Space - lines in the same plane that have no points in common
If two lines intersect, then there is exactly one plane containing them
Skew Lines - lines in space that are neither parallel nor intersecting
If two planes intersect then they intersect in exactly one line
If a line is perpendicular to each of two intersecting lines at their point of intersection then the line is
perpendicular to the plane determined by these lines
Dihedral Angle - union of two half planes with a common edge
Measure of a Dihedral Angle - measure of the plane angle formed by two rays each in different half
planes of the angle and each perpendicular to the common edge at the same point of the edge
Perpendicular Planes - two planes that intersect to form a right dihedral angle
A line is perpendicular to a plane if and only if it is perpendicular to each line in the plane through the
intersection of the line and the plane
A plane is perpendicular to a plane if the line is perpendicular to the plan
Trough a given point of a line, there can be only one plane perpendicular to the given line
Two planes are perpendicular to the same line if and only if the planes are parallel
Distance between two planes - length of the line segment perpendicular to both planes with an
endpoint on each plane
11 - 2 Surface Area of a Prism
Polyhedron - 3D figure formed by the union of surfaces enclosed by plane figures
Prism - polyhedron in which 2 faces called bases of the prism , are congruent polygons in parallel planes
Faces - portion of the planes enclosed by a plane figure
Edges - intersection of the faces
Vertices - intersection of edges
Lateral Sides - surfaces between corresponding sides of the bases
Lateral Edges - common edges of the lateral sides
Altitude - line segment perpendicular to each edge of the bases with an endpoint on each base
Height - length of the altitude
Right Prism - prism in which lateral sides are all perpendicular to the base and all lateral sides are
rectangles
Parallelpiped - prism that has parallelograms as bases
Rectangular Parallelpiped - parallelpiped that has rectangular bases and lateral edges
Lateral Area - sum of the areas of the lateral sides
Total Surface Area - sum of the lateral area and area of the bases
Prism
Lateral Area
Surface Area
Volume
Bh
Pyramid
1/3Bh
Cone
rhs
L + r2
1/3r2hc
Cylinder
2rh
2rh + 2r2
r2h
Sphere
4r2
4/3r3
Chapter 12: Ratio, Proportion, Similarity
12 - 1 Distance
Distance Formula:
 (x2- x1)2+ (y2 - y1)2
12 - 2 Ratio and Proportions/Proportions Involving Line Segments
Ratio - a/b, a:b
Proportion - equation that states two ratios are equal
In a proportion, the product of the means is equal to the product of the extremes
The Mean Proportional - means of proportional are equal, then each is called mean proportional
between 1st and 4th term
2=8
8 32
8 = mean proportional
2, 32 = extremes
A line segment joining the midpoint of two sides of a triangle is parallel to the third side and its length is ½
the length of the third side
When the ratio of the lengths of the parts of one segment is equal to the ratio of the lengths of the parts
of the other = divided proportionally
12 - 3 Similar Polygons/Proving Triangles Similar
AA Triangle Similarity - Two triangles are similar if two angles of one triangle are congruent to the
corresponding angles of the other
SSS Similarity Theorem - Two triangles are similar if three ratios of corresponding sides are equal
A line that is parallel to one side of a triangle and intersects the other two sides in different points cuts off
a triangle similar to the given angle
12 - 4 Proportional Relations Among Segments Related to Triangles
If two triangles are similar the lengths of corresponding altitudes, medians and angle bisectors have the
same ration as the lengths of any two corresponding sides
The perimeters of two similar triangles have the same ratio as the lengths of any pair of corresponding
sides
12 - 5 Concurrence of Medians of a Triangle
Centroid - If 3 medians of a triangle are drawn, they intersect at one point called the centroid
Any two medians of a triangle intersect in a point that divides each median in a ration of 2:1
The medians of a triangle are concurrent
12 - 6 Proportions in a Right Triangle
Projection of a Point on a Line - foot of a perpendicular drawn from the point on a line
Projection of a Segment - when the segment is not perpendicular to the line, the projection of the
segment whose endpoints are projection of the endpoints of the given line segment on the line
The altitude to the hypotenuse of a right triangle divided the triangle into
two triangles that are similar to each other and to the original triangle
The length of each leg of a right triangle is the mean proportional
Between the length of the projection of that leg on the hypotenuse
And the length of the hypotenuse
x
a hyp = b
a=x
x b
The length of the altitude to the hypotenuse of a right triangle is the
mean proportional between the length of the projections of the legs
on the hypotenuse
a alt = x
b
a=x
x b
12 - 7 Pythagorean Theorem
Pythagorean Theorem: a2 + b2 = c2
Chapter 13: Geometry of Circles
13 - 1 Arcs and Angles
All radii of the same circle are congruent
Central Angle - angle whose vertex is the center of the circle
Minor Arc - less than 180
Major Arc - greater than 180
In a circle or congruent circles, central angles are congruent if their arcs are congruent
13 - 2 Arcs and Chords
Chord - line segment whose endpoints are points of the circle
Diameter - chord that has the center of the circle as one of its points
In a circle or in congruent circles, two arcs are congruent if and only if their central angles are congruent
In a circle, two chords are congruent if and only if their arcs are congruent
Apothem - perpendicular segment from the center of the circle to the midpoint of a chord, also length
of segment
Two chords are equidistant from the center of a circle if their distance away is congruent
13 - 3 Inscribed Angles and Their measures
Inscribed Angle - angle whose vertex is on the circle and whose sides are chords
The measure of an inscribed angle of a circle is equal to ½ the measure of its intercepted arc
An angle inscribed in a semicircle is a right circle
If two inscribed angles of a circle intercept the same arc, then they are congruent
13 - 4 Tangents and Secants
Tangent - line in a plane of the circle that intersects the circle in one point
Secant - intersects the circle in two points
A line is tangent to a circle if it is perpendicular to the radius
13 - 5 Angles Formed by Tangents, Chords, and Secants
The measure of an angle formed by a tangent and a chord that intersect at the point of tangency is
equal to ½ the measure of the intercepted arc
The measure of an angle formed by two chords intersecting within a circle is equal to ½ the sum of the
measure of the arcs intercepted by the angles and its vertical angle
The measure of an angle formed by a tangent and a secant, two secants, or two tangents intersection
outside the circle is equal to ½ the difference of the measure of the intercepted arcs
13 - 6 Measure of Tangents Segments, Chords, and Secant Segment
If two chords intersect within a circle, the product of the measure
of the segments of one chord is equal to the product of the measure
of the segment of the other. AE(EB) = CE(ED)
If a tangent and a secant are drawn to a circle from an external
point, the square of the tangent segment equals the product of
the whole segment and its external segment (AB)2 = BD(BC)
A
D
E
C
B
A
D
If two secant segments are drawn to a circle from an external point,
then the product of the lengths of one secant segment and its
external segments is equal to the product of the lengths of the other
secant segments and its external segment AC(AB) = AE(AD)
B
C
C
E
B
D
13 - 7 Circles in the Coordinate Plane
Center-radius equation of a circle:
Center: (h,k)
(x - h)2 + (y - k)2 = r2
Chapter 14: Locus
14 - 1 Five Fundamental Locus
Locus - set of all points, and only those points that satisfy a given set of conditions
1. The locus of points equidistant from 2 fixed points A and B is a perpendicular bisector
.
.
2. The locus of points equidistant from 2 intersecting lines AB and CD
D
A
C
B
3. The locus of points equidistant from 2 parallel lines AB and CD
A
4. The locus of points a fixed distance d from a line AB
5. The locus of points a fixed distance d from fixed point A
.
14 - 2 Parabola
x = -b
2a
 Finds axis of symmetry
Then, plug x-coordinate into the parabola formula to find the y-coordinate
Make a table, using the x from the equation above, and 2 x points below and 2 above
Once Solved, plot all points