Review of Entire Course
POLYGONS
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Sum of interior angles = (number of sides-2) 180= (n-2)180
( n  2)180
One interior angle of a regular polygon =
(n = # of sides)
n
Sum of exterior angles of a polygon = 360 ( no matter how many sides)
360
One exterior angle of a polygon=
n
Properties of a parallelogram: use these for algebra problems and proofs
o Opposite sides parallel
 To prove using coordinate: find slopes of opposite sides and show
that the slopes are equal
o Opposite sides are equal
o Opposite angles are equal
o Consecutive angles are supplementary
o Diagonals bisect each other
o One set of opposite sides are equal and parallel
Properties of a Rectangle: use these for algebra problems and proofs
o Opposite sides parallel
 To prove using coordinate: find slopes of opposite sides and show
that the slopes are equal
o Opposite sides are equal
o Opposite angles are equal
o Consecutive angles are supplementary
o Diagonals bisect each other
o 4 right angles
 to prove using coordinate: show that the slope of two adjacent
sides are negative reciprocals.
o Congruent diagonals
Properties of a Rhombus: use these for algebra problems and proofs
o Opposite sides parallel
 To prove using coordinate: find slopes of opposite sides and show
that the slopes are equal
o Opposite sides are equal
o Opposite angles are equal
o Consecutive angles are supplementary
o Diagonals bisect each other
o Diagonals are perpendicular
o 4 equal sides
 use distance formula 4 times to show all sides are =
o Diagonals bisect opposite angles
o 2 consecutive sides are equal
Properties of a Square: use these for algebra problems and proofs
o Opposite sides parallel
 To prove using coordinate: find slopes of opposite sides and show
that the slopes are equal
o Opposite sides are equal
o Opposite angles are equal
Review of Entire Course
o Consecutive angles are supplementary
o Diagonals bisect each other
o Diagonals are perpendicular
o 4 equal sides
 use distance formula 4 times to show all sides are =
o Diagonals bisect opposite angles
o 4 right angles
 find slope of 2 consecutive sides and show that they are negative
reciprocals and therefore perpendicular
o Congruent diagonals
 Properties of a trapezoid: use these for algebra problems and proofs
o Only one pair of opposite sides parallel
 To prove using coordinate: Find the slope of all four
Sides. Show two sides have = slopes and are parallel and that the other
two sides have unequal slopes and therefore are not parallel.
o Isosceles trapezoid: 2 = legs, = base angles.
 To prove using coordinate: Find the slope of all four
Sides. Show two sides have = slopes and are parallel and that the other
two sides have unequal slopes and therefore are not parallel. Also, find
the distance of the two non-parallel sides and show they have = length.
o Congruent diagonals
Coordinate
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PARALLEL LINES have EQUAL SLOPES
y  y1
o Slope formula: m  2
x 2  x1
PERPENDICULAR LINES have NEGATIVE RECIPROCAL SLOPES
2
3
o Examples of negative reciprocals: and
3
2
o Two lines with a slope of 0 and undefined are considered negative
reciprocals and are therefore perpendicular
General equation of a line: y = mx+b
o m is the slope and b is the y-intercept
o If they give you a point that is on the line, plug in the coordinate (x,y) into
y = mx+b and plug in the slope. Then solve for b.
o if they give you two points first find the slope using the slope formula,
then pick one of the points (x,y) and plug it into y = mx+b and plug in the
slope. Then solve for b .
Midpoint formula:
o Use to show that two segments bisect each other by showing that they
have the same midpoint.
 x  x 2 y1  y 2 
,
 Midpoint=  1

2 
 2
Distance Formula
o Used to find the length of a segment
o Distance =
( x 2  x1 ) 2  ( y 2  y1 ) 2
Review of Entire Course
Logic
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Negate: add the word not to the statement (or take the word not out)
Inverse: negate both statements
o Example: if it rains, then I bring an umbrella. Inverse: If it does not rain
then I did not bring an umbrella.
Converse: Switch the order
o Example: Converse: If I bring an umbrella then it rains
Contra positive: Switch and negate
o Example: If I do not bring an umbrella then it is not raining.
o LOGICALLY EQUIVALENT: have the same truth value.
Disjunction:
o “OR”, example: It is raining or it is sunny
o Only False is both statements are False
Conjunction:
o “AND” , example: 2 is even and 3 is odd
o Only True if both statements are True
Conditional
o “IF”  “THEN”, example: If it snows then it is cold.
o Only False if T F
Biconditional
o “IF AND ONLY IF”, example: Two lines are parallel if and only if they
never intersect.
o True if both statements are True, True if both statements are False.
Transformations
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Isometry: size doesn’t change
Reflection: notation: rx-axis. Line reflections do not preserve orientation.
Opposite or reverse or indirect isometry
o rx-axis : Reflect over the x-axis: switch the sign of the y value
 Example: (7,-3) goes to (7,3)
o
r
: Reflect over the y-axis: switch the sign of the x value
y-axis
Line reflections
 Example: (7,-3) goes to (-7,-3)
o ry=x: Reflect over the line y=x: flip flop the x and y value.
 Example: (7,-3) goes to (-3,7)
o
r
Reflection through the origin: switch the sign of both x and y
origin:
Point reflection
 Example: (7,-3) goes to (-7,3)
 Rotation: Spin your paper counterclockwise (left!!). Notation: R90. Is a direct
isometry
o R90: moves one quadrant to the left (counterclockwise)
 Rule: (x,y) goes to (-y,x)
 Example: (7,-3) goes to (3,7)
o R180: moves two quadrants to the left (counterclockwise)
 Rule: (x,y) goes to (-x,-y) same as Reflection through the origin
 Example: (7,-3) goes to (-7,3)
o R270: moves three quadrants to the left (counterclockwise)
 Rule: (x,y) goes to (y,-x)
 Example: (7, -3) goes to (-3,-7)
Review of Entire Course
 Dilation: Changes size. Notation Dk , Is not an isometry
o k is called a scale factor
o multiply both the x and y value of a point by k
 Example: D2 on the point (7,-1) becomes (14,-2)
 Translation: Is a slide. Only the location of the points changes. Is a direct
isometry?
o Notation: T(a,b), where you add a to the x value and b to the y value
o You add values to the x and y coordinates.
 Example: T(-2,5), on the point (1,-3). New point: (-2+1, 5+-3) =
(-1,2)
 Composition of Transformations: Read from RIGHT to LEFT
o Example: ry  axis  T 2, 4 (7,3)
 First do the Translation: (7+-2, -3+4) = (5,1)
 Now Reflect the NEW POINT over the y-axis. (5,1) goes to (-5,1)
 Glide reflection
o A glide reflection is the composition of a translation and a line reflection.
o Order does not matter.
o The Translation has to be parallel to the line of reflection
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Helpful hints:
o If a question says which of the following is not an isometry: look for the
one with Dilation
o If a question asks which of the following does not preserve orientation or
is an indirect isometry look for the answer with some sort of line
reflection. (this does not include reflection through the origin since the
origin is not a line)
LOCUS
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EQUIDISTANT means THINK LOCUS
d
A fixed distance from A POINT is a CIRCLE.
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A fixed distance from A LINE is PARALLEL LINES
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Equidistant from TWO PARALLEL LINES is a 3RD PARALLEL LINE
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Equidistant from TWO POINTS is a PERPENDICULAR BISECTOR
d
d
Review of Entire Course
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Equidistant from INTERSECTING LINES is ANGLE BISECTORS
Compound locus: draw each locus separately and put an X where the dotted lines
cross
CIRCLES
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Equation of a circle: ( x  h) 2  ( y  k ) 2  r 2
o (h,k) is the center of the circle. r is the radius.
o When given the equation, flip the sign of the numbers to find the center
and square root to find the radius.
 Example: ( x  2) 2  ( y  3) 2  25 . Center is (2,-3) and radius is 5
ANGLES
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Central Angle: An angle whose vertex is the center of the circle
o RULE: Central Angle= measure of the intercepted arc
 Example: AOB = m arc AB, therefore AOB= 30
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Inscribed Angle: An angle whose vertex is on the circle.
o RULE: Inscribed angle = one half the measure of the intercepted arc
 Example: m arc AC= 60, therefore ABC= (1/2) 60= 30
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30
60
Tangent Chord Angle: an angle formed by a tangent and a chord or a tangent
and a diameter.
o RULE: Angle= (1/2) intercepted arc
 Example: ABD= ½ m arc BD
100
ABD= ½ (100)= 50
A
B
D
C
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Angles inside the circle formed by intersecting chords: these angles are on the
inside of the circle. A chord is a line in the circle that does not go through the
center.
50
o RULE: Angle= ½( the sum of the intercepted arcs)
 Example: AEC= ½ (arcAC+ arcDB)
70
AEC= ½ (70+ 50)= ½ (120)= 60
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Angles outside the circle formed by 2 tangents, 2 secants or a secant and a
tangent: a secant starts outside the circle and intersects the circle 2 times, a
tangent starts outside of a circle and intersects the circle 1 time.
o RULE: outside angle = ½( the difference of the intercepted arcs)
60
 Example: ACE = ½ (arcAE - arcDB)
ACE = ½ (60-20)= ½ (40) = 20
SEGMENTS
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Intersecting chords: segments are all on the inside of the circle.
o RULE: part x part = part x part
 Example: AE  EB  CE  ED
x
4
3
6
20
Review of Entire Course
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x6  43
6x = 12, so x =2
Two secants: 2 secants that intersect outside of the circle
o RULE: whole secant x outside = whole secant x outside
o Example: AC  BC  CE  CD
12  2  ( x  4)  4
24  4 x  16
10
2
x
4
8  4x
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2x
A secant and a tangent: a secant and a tangent intersect outside of the circle
o RULE: tangent2 = whole secant x outside
9
2
 Example: GH  JH  IH
3
x2= 12(3)=36
x
x=6
Two tangents: two tangents drawn from the same external point are equal
o RULE: LM  LK
Diameter perpendicular to a chord: we know many things when this happens
o RULES:
arc KH= arc IH
arc KJ= arc JI
*KL= LI
TO USE FOR CIRCLE PROOFS
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Central angle = the measure of the intercepted arc
Inscribed angle= one half the intercepted arc
Parallel Chords: parallel lines intercept = arcs
 Example: arc AD = arc BC
A
B
C
D
= chords intercept = arcs, and = arcs intercept = chords
o Given: arc KL= arc MN then MN=KL
o Given KL= MN then arc MN= arc KL
Inscribed angles that intercept the same arc are =
A
o  DAB =  DCB, and  ADC =  ABC
Inscribed angles that intercept = arcs are =
C
B
O
TRIANGLES
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D
Sum of interior angles in a triangle is 180
Triangle Inequality: the sum of any two sides of a triangle needs to be longer
than the third side.
B
o RULE: AB+BC> AC
BC+AC> AB
AB+AC>BC
A
C
Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of
the two non-adjacent interior angles.
c
o RULE: a = b + c
b
a
Review of Entire Course
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Altitude Rule and Leg Rule: Use when you see the following pictures
Whole Hypotenuse
Seg1
alt
Seg 2
Altitude Rule:

alt
Seg 2
Seg 1
Leg 2
WholeHyp . leg #
Altitude
Leg Rule:

leg #
seg #
Leg 1
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Types of Triangles:
o Isosceles Triangle: 2 = angles across from 2 = sides
o Equilateral Triangle: 3 = sides, 3 = angles (all 60)
o Scalene Triangle: all 3 sides are NOT =
o Obtuse Triangle: a triangle with ONE obtuse angle and 2 acute
angles
o Acute Triangle: ALL THREE angles are acute. (< 90)
Pythagorean Theorem
o a 2  b 2  c 2 , where c has to be the longest side (the hypotenuse)
o only works for a right triangle
The Longest Side is across from the largest angle.
The exterior angle of a triangle is always greater than either of the two nonadjacent interior angles
Similar Triangles: Set up a proportion and cross multiply.
o All angles are equal but sides are proportional
o Proofs:
 Level 1: Prove ABC ~ DBE, Use: Angle-Angle
AB BC

 Level 2: Prove:
, Use: Defintion of Similar
DE BE
Polygons. Note: Prove Level 1 FIRST
 Level 3: Prove AB(BE) = (BC)(DE), Use: Means Extremes
Note: Prove Level 1 and 2 FIRST
ANGLES
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Supplementary Angles: 2 angles whose sum in 180
Complementary Angles: 2 angles whose sum is 90
Acute Angle: an angle less than 90
Obtuse Angle: an angle greater than 90
Linear Pair: 2 angles that form a line
Review of Entire Course
PARALLEL LINES CUT BY A TRANSVERSAL
 Alternate interior angles: equal: 4 & 6 and 5&3
 Corresponding angles: equal: 2 & 5, 4 & 7, 1 & 6, and 3 & 8
A
 Same side interior: supplementary: 4 & 5, and 3 & 6
 Alternate Exterior angles: equal: 2 & 8 and 1 & 7
 Same side Exterior angles: supplementary: 2&7 and 1 & 8 C
t
2
4
5
7
SOLIDS
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Area
o Rectangle: A= base x height
o Triangle: A= ½ x base x height
o Square: A= side x side
o Trapezoid: A= ½ x (base1 +base2) x height
Volume
o Rectangular Prism: V= length x width x height
o Cube: V= side x side x side
o Triangular Prism: V= (1/2 x base x height) x Height of Prism
o Cylinder: V= Bh= r2h
o Cone: V= (1/3)Bh= (1/3) r2h
o Pyramid: V= (1/3)Bh= (1/3) side of base x side of base x altitude
o Sphere: (4/3) r3
Surface Area
o Prism: Find the area of all sides and add them together
o Cylinder: SA= 2r2+ 2rh
o Cone: SA= r2 + rl (l is the slant height)
o Sphere: SA= 4r2
Lateral Area
o Always refer to reference sheet on exam.
o Right Circular Cone: L  rl , where l is the slant height
o Right Circular Cylinder: L  2rh
PLANES
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B
6
8
TO USE FOR PROOFS
 Alternate interior angles on parallel lines are equal
o Highlight the parallel lines and the transversal and look for the “Z”
 Corresponding angles on parallel lines are equal
o Highlight the parallel lines and the transversal and look for the “F”
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1
3
2 planes intersect to form a line
There are four ways to form a plane
o Intersecting lines, parallel lines, three non-collinear points, a line and a
point not on the line
Coplanar: points or lines that are on the same plane
Collinear: points that are on the same line
D
Review of Entire Course
 There is one plane that can be made that is perpendicular to a point on a line
 A plane that intersects two parallel planes forms two parallel lines.
 Two lines that are perpendicular to the same plane and parallel and coplanar
 There are an infinite number of planes that are perpendicular to a plane and go
through a point not on the plane
 There is only one line that can be perpendicular to a plane that goes through a
point not on the plane.
PROOFS
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3 types of proofs: Congruent Triangle Only, CPCTC and Indirect
o Congruent Triangle only: Prove statement looks like: ABCDEF
 Last reason will be: SAS, ASA, AAS, SSS or HL
 NO ASS OR SSA
 HL only works for right triangles
o CPCTC: When proving parts, like angles or segments. Prove statement
may look like: AB CD or ABC  DEF
 CPCTC COMES AFTER PROVING TWO TRIANGLES ARE
CONGRUENT
 Use CPCTC when proving lines are parallel
The keyword from the statement helps you find the reason for the next statement
SEE PROOF CHART
BUILD PROOFS
o Use when they give you pieces and you have to put them together to make
bigger segments
o Pieces ---> Wholes
 Given
 Reflexive Axiom (only if needed)
 Addition Axiom
 Partition Axiom
 Substitution Axiom
CHOP PROOFS
o Use when they give you longer pieces and you have to chop some of it off
to get the segment that you want
o Wholes ---> Pieces
 Given
 Partition Axiom
 Substitution Axiom
 Reflexive or Given
 Subtraction Axiom
Indirect Proofs
o Step 1: Assume the opposite of the prove statement.
o Save the Given with the not in it (or the one that talks about the type of
triangle) for the 2nd to last step
o Complete the proof just like you would if it were a regular proof
o Last step is contradiction
Review of Entire Course
Misc. Topics
Name
Picture
Description

Located at intersection of the angle bisectors.
Incenter
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Circumcenter
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Located at intersection of the perpendicular
bisectors of the sides.
For an obtuse triangle the circumcenter is located
outside the triangle
 Located at intersection of medians.
 Always located inside the triangle
 The centroid is two-thirds the way along each
median
 The centroid divides each median into two
segments whose lengths are in the ratio 2: 1, the
longest segment is near the vertex
Centroid
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Orthocenter
Located at intersection of the altitudes of the
triangle.
For an obtuse triangle the orthocenter is located
outside the triangle
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Mid-Segment of a Triangle
When you connect the midpoints of two sides of a triangle it is called the mid-segment.
Mid-segments have the following properties:
1. The mid-segment of a triangle joins the midpoints of two sides of a triangle such
that it is parallel to the third side of the triangle.
2. The mid-segment of a triangle joins the midpoints of two sides of a triangle such
that its length is half the length of the third side of the triangle.
Given: D is the midpoint of AC. E is the midpoint
of BC. Mid-Segment DE
Therefore:
DE || AB and DE = ½ AB

Mid-Segment of a Trapezoid
When you connect the midpoints of the two legs of a trapezoid it is called the mid-segment
The mid-segment of a Trapezoid has the following properties:
1. The mid-segment is parallel to both bases.
2. The mid-segment has length equal to the average of the length of the
bases.
Review of Entire Course
CONSTRUCTIONS
Perpendicular Bisector
Perpendicular at a point on a line
Perpendicular to a line through an
external point
Bisect an Angle
Parallel Lines through a point
Constructing A Median in a Triangle