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Name:_______________________
Final/Regents Review Packet
Date:_____ Period:____
Mr. Woods
Final/Regents Review Packet
Some Key Vocabulary:
 Orthocenter of a Triangle: The point of intersection of the three altitudes of a
triangle.
 Incenter of a Triangle: The point of intersection of the three angle bisectors of a
triangle. It is also the center of the inscribed circle of a triangle.
 Circumcenter of a Triangle: The point of intersection of the three perpendicular
bisectors of a triangle. It is also the center of the circle that can be circumscribed
about a triangle.
 Centroid of a Triangle: The point of intersection of the three medians of a triangle.
 Collinear Points: Points that are on the same line
 Parallel Lines: Lines that never intersect. Parallel lines have the same slope.
 Perpendicular Lines: Lines that intersect to make a 90˚ angle. Parallel lines have
negative reciprocal slopes.
 Complementary Angles: Angles that add up to 90˚.
 Supplementary Angles: Angles that add up to 180˚.
 Linear Pair: Two angles that make up a line.
Important Geometric Relationships:
 Vertical angles are congruent.

p || q If two parallel lines are cut by a transversal, any two of the eight angles that
are formed are either congruent or supplementary.
Angles in Triangles:
 The sum of all of the angles in a triangle add up to 180˚
 The exterior angle of a triangle is equal to the sum of the non-adjacent interior
angles
 The side opposite the largest angle of a triangle is the longest side.
 The side opposite the smallest angle of a triangle is the smallest side.
 When two sides of a triangle are congruent, the angles opposite them are also
congruent.
Triangle Inequality Theorem:
The sum of the two shortest sides of a triangle are greater than the third side.
Proving Triangles Congruent:
There are five ways to prove triangles congruent
 ASA
 SAS
 AAS
 SSS
 HL
***You cannot prove triangles congruent using SSA or AAA***
Proving Triangles Similar:
Two triangles are similar if any one of the following is true:
 AA
 Corresponding sides are in proportion
 The lengths of two pairs of sides are in proportion and their included angles are
congruent
Angles of a Polygon:
In a Polygon with n sides:
 Sum of the exterior angles = 360˚
 Sum of the interior angles = 180(n-2)
In a Regular Polygon with n sides:
 Each exterior angle = 360/n
 Each interior angle = 180(n-2)/n
Properties of Parallelograms:
In a parallelogram:
 Opposite sides are parallel
 Opposite sides and opposite angles are congruent
 Diagonals bisect each other
In a rectangle:
 All the properties of a parallelogram
 Four right angles
 Congruent diagonals
In a Rhombus:
 All the properties of a parallelogram
 Four congruent sides
 Diagonals bisect opposite angles
 Diagonals are perpendicular
In a Square:
 All the properties of a parallelogram
 Four right angles
 Four congruent sides
 Congruent diagonals
 Diagonals are perpendicular
 Diagonals bisect opposite angles
In an Isosceles Trapezoid:
 Only one pair of opposite sides parallel
 Congruent diagonals
Coordinate Geometry Formulas:
 Midpoint: ((x1 + x2)/2, (y1 + y2)/2)
 Use to prove that segments bisect
 Slope: (y1-y2)/(x1-x2)
 Use to prove that segments are parallel or perpendicular
 Distance: √( x1-x2)2 + (y1-y2)2
 Use to prove that segments are congruent
Right Triangle Relationships:
Altitude Rule:
part of hypotenuse
=
altitude
altitude
other part of hypotenuse
Leg Rule:
hypotenuse
leg
=
leg
leg
leg
projection
projection
projection
“Special” Right Triangles:
Midsegment Theorems:
Midsegment of a Triangle:
 Parallel to the base
 Half the length of the base of triangle
 The perimeter of a triangle formed by the midsegments, is one half the perimeter
of the larger triangle.
Midsegment of a Trapezoid:
 Parallel to the two bases of a trapezoid
 One half the length of the sum of the bases
Centroid Relationships:
**Remember: The centroid is the intersection of the three medians of a triangle***
B
F
A




P
D
E
C
AD, CF, and BE are all medians of the triangle.
They are concurrent at point P
The three medians divide the triangle into six regions of equal area
Known as the “center of gravity” of a triangle
Also the centroid divides the medians into two segments in the ratio
AP 2

PD 1
and
CP 2

PF 1
and
2
, such that
1
BP 2

PE 1
(remember: the top of the ratio are the parts of the median from the
vertices to the centroid or the longer segment is near the vertex)
The centroid is exactly two-thirds the way along each median
AP BP CP 2



AD BE CF 3
2
(AD) = AP
3
Notice that AP, BP, and CP are the parts
2
(BE) = BP
3
of the medians drawn from the vertex
to the centroid
2
(CF) = CP
3
The centroid is located one-third of the perpendicular distance between each side and the opposing
point
PD PE PF 1



AD BE CF 3
1
(AD) = PD
3
1
(BE) = PE
3
1
(CF) = PF
3
Area






Formulas:
Triangle:
Parallelogram:
Rectangle:
Rhombus:
Square:
Trapezoid:
A = ½bh
A = bh
A = lw
A = ½ d1d2 (d = diagonal)
A = s2 or A = ½ d2
A = ½ h(b1 + b2)
Surface Area Formulas:
 Cube:
S.A. = 6e2
 Rectangular Prism:
S.A. = 2(lw + lh + hw)
 Any prism:
S.A. = Lateral Area + Area of two ends
 Sphere:
S.A. = 4πr2
 Cylinder:
S.A. = 2πr2 + 2πrh
***Tip: Make sure to keep your units straight. Surface area should always be in square
units***
Lateral Area Formulas:
 Prism:
 Pyramid:
 Cylinder:
 Cone:
L.A. = (base perimeter)h
L.A. = ½ (base perimeter)(slant height)
L.A. = 2πrh
L.A. = πr(slant height)
Volume Formulas:
 Rectangular Prism:
 Any Prism:
 Pyramid:
 Cylinder:
 Cone:
 Sphere:
V = lwh
V = (area of base)h
V = 1/3 (area of base)h
V = πr2h
V = 1/3 πr2h
V = 4/3 πr3
General Equations:
 Slope-intercept equation of a line: y=mx+b, where m is the slope and b is the y-int
o Use this equation when graphing a line
o To graph: start at the y-intercept and then either go up down depending on
the slope
o Positive Slope: Up and to the right
o Negative Slope: Down and to the right
 Point-slope equation of a line: y – y1 = m(x – x1), where (x1,y1) is a point and m is slope
o Use this equation when you are asked to find the slope of a line and given a
point and a slope.
o To graph: must then put this equation into slope-intercept form
 Circle with center (h,k) and radius r: (x – h)2 + (y – k)2 = r2
o Use this equation when graphing a circle.
o First graph the center, and then count up for the radius, count to the left,
count down, and count to the right. Sketch in a circle graph connecting all
four of those points. (try the best you can to make a good circle! )
Circle Relationships:
Arcs and Angles:
 Central Angle: An angle formed by two radii of a circle. This angle comes from the
center of the circle. Here, the angle is equal to the intercepted arc.
 Inscribed Angle: An angle that is formed by two chords and is on the circle. Here,
the angle is equal to one-half the intercepted arc.
 Tangent/Chord Angle: An angle that is formed by a tangent and a chord. This angle
is on the circle. Here, the angle is equal t one-half the intercepted arc.
 Angle Inside the Circle: This angle is formed by two chords. It makes a weird “plus”
inside the circle. Here, you add together the two intercepted arcs and then divide
by two to get the angle. (hint: look in front of the angle and then look behind the
angle to get the two arcs).
 Angle Outside the Circle: This angle is either formed by two tangents, a tangent
and a secant, or two secants. To find the angle, big arc – small arc and then divide
by two.
***Hint: If you are given an angle and asked to find an arc – first look at where the
vertex of the angle is in relation to the circle. Then, write out the formula. Solve for
the arc that way.***
***You can never combine angles and arcs. So do not do it in these problems!!!***
Tangent/Secant/Chord Segments:
 Chords and Circles
o In a circle a radius (or a diameter) that is perpendicular to a chord bisects
the chord
o In a circle, if two chords are congruent then their intercepted arcs are also
congruent.
o In a circle, if two chords are parallel, then the arcs in between them are
congruent
o Two Intersecting Chords in a Circle:
 (segment piece) x (segment piece)=(segment piece) x (segment piece)
Example: CE x ED = AE x EB

Tangent and a Secant:
o (tangent)2 = (whole)(outside)
Example: PA2 = (PB)(PC)

Two Intersecting Secants:
o (whole)(outside) = (whole)(outside)
Example: (PB)(PA) = (PD)(PC)

Two Tangents that meet at the same external point:
o tangent = tangent
Example: (PA) = (PC)
Transformations:
 Isometry: A transformation that preserves length
Transformation
Properties
Preserved
Isometry
Coordinate Rules
Line Reflection
Collinearity
Angle Measure
Distance
Opposite
rx-axis(x,y) = (x,-y)
ry-axis(x,y) = (-x,y)
rorigin(x,y) = (-x,-y)
ry=x(x,y) = (y, x)
ry=-x(x,y) = (-y, -x)
rx=k(x,y) = (2k - x,y)
ry=k(x,y) = (x, 2k - y)
Translation
Collinearity
Angle Measure
Distance
Orientation
Direct
Ta,b(x,y) = (x+a, y+b)
Rotation
*always counterclockwise unless
otherwise noted*
Collinearity
Angle Measure
Distance
Orientation
Direct
Dilation
Collinearity
Angle Measure
Orientation
NOT
Glide Reflection
Collinearity
Angle Measure
Distance
Opposite
R90˚(x,y) = (-y, x)
R180˚(x,y) = (-x, -y)
R270˚(x,y) = (y, -x)
R-90˚(x,y) = (y, -x)
R-180˚(x,y) = (-x, -y)
R-270˚(x,y) = (-y, x)
Dk(x,y) = (kx, ky)
*where k is the
“scale factor”
Follow the
coordinate rules of
the given
transformations
Composition Transformation:
When two or more transformations are combined to form a new transformation, the result
is called a composition of transformations. In a composition, the first transformation
produces an image upon which the second transformation is then performed.
The symbol for composition is an open circle.
The notation rx-axis ◦ T3,4 is read as a reflection in the x-axis FOLLOWING a translation of
(x+3,y+4). Basically, this process is done in REVERSE. So whenever you see a composition
of a transformation, you work BACKWARDS!!!
Triangle Proofs:
 Always first mark up your diagram to show what segments and angles are congruent
 First list the information that is given to you first
 Then, see what you can get from that given information
 Next, look at your diagram and see if there is anything in the picture to use to
prove that sides or angles are congruent. Look for:
o Reflexive Property: In the reflexive property, the triangles share something
– either a side or an angle.
o Vertical Angles: Angles that “look” at each other. Vertical angles are always
congruent
 Once you are done, use one of the 5-ways to prove triangles congruent (see earlier
in the packet)
 If you are asked to prove segments congruent or angles congruent, look at the
diagram and see which two triangles you first want to prove congruent
o C.P.C.T.C = Corresponding Parts of Congruent Triangles are Congruent
Locus: (a bunch of points that all do the same thing)
 Locus of a point: a circle around that point.
 Points equidistant from Two Points: a line between those two points; perpendicular
bisector.
 Locus a fixed distance from a line: two parallel lines, one on either side.
 Equidistant from two parallel lines: a parallel line midway between them.
 Equidistant from two Intersecting lines: bisectors of the vertical angles ; form a
right angle to each other.
Logic:
 Inverse: Negate both statements in the conditional
 Converse: Switch the statements in the conditional
 Contrapositive: Switch and negate the statements in the conditional. The
contrapositive is logically equivalent to the conditional.
Please Note: This packet just acts as a guide for what to study. Please review any other
packets that I give you. (Especially the one on constructions). In addition, you will still
need to go over your Final Review Packet and your Green Book. Again, if you have any
questions throughout the course of your studying, please feel free to e-mail me.
Good luck on your final and your Regents Exam!!! 