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Transcript
REVIEW PACKET
POLYGONS
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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: find slope of all sides, show opposite
sides have = slopes and that adjacent sides have negative reciprocal
slopes.
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
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
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 =
REVIEW PACKET

o Diagonals bisect opposite angles
o 4 right angles
o Congruent diagonals
Properties of a trapezoid: use these for algebra problems and proofs
o Only one pair of opposite sides parallel
o Isosceles trapezoid: 2 = legs, 2 = base angles
 Congruent diagonals
Logic
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Inverse: negate both statements
o Example: if it rains, then I bring an umbrella. Inverse: If I do not bring an
umbrella then I it did not rain.
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
Transformations
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Isometry: size doesn’t change
Reflection: notation: rx-axis. Does 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 ry-axis: Reflect over the y-axis: switch the sign of the x value
 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 rorigin: Reflection through the origin: switch the sign of both x and y
 Example: (7,-3) goes to (-7,3)
Rotation: Spin your paper counterclockwise. Notation: R90. Is a direct isometry
o R90: moves one box to the left (counterclockwise)
 Rule: (x,y) goes to (-y,x)
 Example: (7,-3) goes to (3,7)
o R180: moves two boxes 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 boxes to the left (counterclockwise)
 Rule: (x,y) goes to (y,-x)
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.
REVIEW PACKET
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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)
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)
CIRCLES

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
TRIANGLES
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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.
o RULE: a = b + c
c
b

a
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 =
REVIEW PACKET
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o Obtuse Triangle: a triangle with ONE obtuse angle and 2 acute
angles
o Acute Triangle: ALL THREE angles are acute.
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
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
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 & 7 and 1 & 8
 Same side Exterior angles: supplementary: 2&7 and 1 & 8 C
t
2
4
5
7
1
3
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”
PROOFS

B
 3 types of proofs: Congruent Triangle Only, CPCTC
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 triangle
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
The keyword from the statement helps you find the reason for the next statement
D
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