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Geometry EOC Study guide
Benchmarks:
Be able to find the converse, inverse, and
contrapositive of a statement.
Determine if two propositions are logically
equivalent
Find the lengths and midpoints of line
segments in two-dimensional coordinate
system
Use coordinate geometry to find slopes of
parallel lines, perpendicular lines, and
equations of lines.
Examples:
If today is Sunday, then tomorrow is
Monday
 If tomorrow is Monday , then today is
Sunday is the _________________ of the
conditional statement
 If today is not Sunday, then tomorrow
is not Monday is the __________________
of the conditional statement
 It today is not Monday, then
tomorrow is not Sunday is the
___________________ of the conditional
statement.
These two statements are logically
equivalent:
 If Lisa is in France, then she is in
Europe
 If Lisa is not in Europe, then she is not
in France
The second statement is the
_______________________ of the first.
Midpoint: the point on a line segment that is
the same distance from both endpoints.
The midpoint ______________ a line segment.
Midpoint Formula:
Slope formula:
Slope intercept form of a line:
__________________________ lines have the same
slope.
___________________________lines have opposite
reciprocal slopes.
To determine the slope of the line 2y = 3x -6
you need to isolate the ___ variable.
 The slope of the line 2y = 3x – 6 is ____
 The line -6x + 4y = 15 is ______________
Identify and use the relationships between
special pairs of angles formed by parallel
lines and transversals.
to the line 2y = 3x – 6
 The line -10x – 15y = 4 is ______________
to the line 2y = 3x - 6
Corresponding angles (draw a picture
Alternate interior angles (draw a picture)
Alternate exterior angles(draw a picture)
Identify and describe convex, concave,
regular, and irregular polygons.
The converse of the parallel line conjecture
states if two lines are cut by a transversal to
form pairs of congruent _________________
angles, congruent ____________________________
angles, or congruent _________________________
Angles, then the lines are _____________________.
Convex polygons:
No diagonals will fall outside the polygon
Concave polygons:
At least one diagonal is outside the polygon
Regular polygons:
All angles and sides are congruent
Irregular polygon:
Does not have all angles or all sides
congruent
Determine the measures of interior and
exterior angles of polygons, justifying the
method used.
The sum of the exterior angles of ANY
polygon is always _________ degrees
Formula to find the sum of the interior
angles in a polygon:
Formula to find the measure of one angle in
a regular polygon:
Use properties of congruent and similar
polygons to solve mathematical or real
work problems.
Apply transformations to polygons to
determine congruence, similarity, and
symmetry. Know the images formed by
translations, reflections, and rotations are
congruent to the original shape.
Remember that before you can solve a math
problem using proportions or set any parts
of polygons equal to each other you need to
first prove the two figures congruent or
similar.
Triangle congruence shortcuts:
 SSS
 SAS
 ASA
 AAS
Triangle similarity shortcuts:
 AA
 SSS
 SAS
Dilations of polygons show
____________________ , not _____________________.
 A dilation either enlarges or shrinks a
polygon by a scale factor that changes
all values of every coordinate of all
vertices
Translation is done by sliding a figure in a
straight line either up or down. (figures will
remain congruent b/c size or shape does
not change)
 When a figure moves to the right by
adding a constant number the only
values changing are the ____ values
 When a figure moves down by
subtracting a constant number the
only values changing are the ____
values.
Reflections produce a figures mirror image
(figures will also remain congruent)
 Reflection about y-axis (x,y) > (-x,y)
 Reflection about x-axis (x,y) > (x,-y)
 Reflection about the line y=x (x,y) >
(y,x)
Know the difference between area and
perimeter of polygons
Rotations are when all points in the original
figure rotate an identical number of degrees
about a fixed point. (figures will remain
congruent)
 My suggestion to keep rotations as
simple as possible is the use easier
points that fall on the x or y axis since
we know that each quadrant on the
coordinate plane is a 90˚ angle
 90 ˚ rotation counterclockwise (x,y) >
(-y,x)
 90˚ rotation clockwise (x,y) > (y,-x)
Area formulas:
Rectangle/Square/Parallelograms:
Triangles:
Rhombuses/kites:
Trapezoids:
Circles:
Circumference of a circle:
Regular polygons:
Determine how changes in dimensions
affect the perimeter and area of common
To determine how two figures compare in
volume or surface area, pick a number to
geometric figures.
start with and find the surface area or
volume.
Then change the original number however
the problem is telling you to and find the
new SA or volume and compare both
answers.
Example: Kendra has a compost box that
has a shape of a cube. She wants to increase
the size of the box by extending every edge
of the box by half of its original length. After
the box is increased in size, what % of the
volume of the new box is the old box?
Describe, classify, and compare
relationships among quadrilaterals
including the square, rectangle, rhombus,
parallelogram, trapezoid, and kite.
Quadrilaterals: polygon with 4 sides
Trapezoid: a quadrilateral with _____ pair of
parallel sides
 The ____________________ angles are
supplementary
 A trapezoid is a/n _____________________
trapezoid if the pair of
___________________________ sides are
_________________.
 Base angles of a/n ____________________
trapezoid are ________________________.
 Diagonals of a/n ______________________
trapezoid are _________________________.
Kites: a quadrilateral with two pairs of
consecutive sides congruent
 Non vertex angles are __________________
 Diagonals are perpendicular
 The diagonal connecting the vertex
angles is the perpendicular
___________________of the other diagonal.
 The vertex angles are bisected by a
Use coordinate geometry to prove
properties of congruent, regular, and
similar quadrilaterals.
diagonal.
Parallelograms: a quadrilateral with 2 pair
of parallel sides.
 Opposite angles are ____________________.
 Consecutive angles are
___________________.
 Opposite sides are _____________________.
 Diagonals of a parallelogram
________________ each other.
Rectangle: (subset of a parallelogram) have
all angles congruent. (four 90˚ angles)
 All 5 properties of parallelograms
apply to rectangles in addition to the
fact that diagonals of a rectangle not
only bisect each other but are also
____________________.
Rhombus: (subset of a parallelogram) have
all sides congruent.
 All 5 properties of parallelograms also
apply to rhombuses in addition to the
fact that diagonals of a rhombus are
perpendicular to each other as well as
bisect each other.
 Diagonals of a rhombus also bisect the
angles of the rhombus.
Square: A regular parallelogram (the baby
of rectangle and a rhombus)
 The diagonals of a square are
_____________________, _____________________,
and __________________ each other.
 A square has all properties of
parallelograms, rectangles, and
rhombuses.
When in the coordinate plane you need to
use the distance formula to determine if all
sides are congruent. (Remember that if the
lines are not vertical or horizontal you
cannot count the squares!!!!)
You also need to find slopes of opposite
sides to determine if they are parallel and
slopes of consecutive sides to determine if
they are perpendicular.
 Remember parallel lines have the
Prove theorems involving quadrilaterals
Classify, construct and describe triangles
that are right, acute, obtuse, scalene,
isosceles, equilateral, and equiangular.
Define, identify, and construct altitudes,
medians, angle bisectors, perpendicular
bisectors, orthocenter, centroid, incenter,
and circumcenter.
same slope
 Perpendicular lines have opposite
reciprocal slopes.
See page 298 and review a view examples
Right triangle: Triangle with a 90˚ angle.
 The side opposite the right angle is
called the __________________________
 The other two sides are called ________
 You can only use sine, cosine, and
tangent as well as the Pythagorean
Theorem in right triangles!!!!!!
 Remember not to assume a triangle is
right if it is not stated in the problem,
if no angles are drawn
Acute: a triangle with all angles less than
90˚
Obtuse: one angle in a triangle is greater
than 90˚
 Remember that there can only be one
angle greater than 90˚ in a triangle
Scalene: no sides are congruent
Isosceles: two sides congruent
Equilateral: all sides congruent
Equiangular: all angles congruent
 Equilateral triangles are regular
polygons whose three angles each
measure 60˚
 Remember that the height of a
triangle has to be perpendicular to its
base. In an equilateral or isosceles
triangle you need to find the height by
bisecting the vertex angle which then
becomes the perpendicular bisector
of the base.
Altitude: the perpendicular segment from a
vertex to its opposite side
 The _______________________is the point of
concurrency for the altitudes
Medians: the segment that connects a
vertex to the midpoint of its opposite side
 The _____________________ is the point of
concurrency of the medians.
Prove that triangles are congruent or
similar and use the concept of
corresponding parts of congruent triangles
(CPCTC)
Apply the inequality theorems
Angle bisectors: the segment that cuts an
angle in half
 The ____________________ is the point of
concurrency that connects the angles
bisectors
Perpendicular bisectors: the line that
bisects and is perpendicular to each side of
the triangle
 The ___________________________ is the
point of concurrency of the
perpendicular bisectors.
Remember that BEFORE you can prove
parts congruent by CPCTC you first HAVE to
prove triangles congruent by SSS, SAS, ASA,
or AAS.
Triangle inequality conjecture: the sum of
the lengths of any two sides of a triangle is
greater than the length of the third side.
Side-Angle inequality conjecture: in a
triangle if one side is longer than another
side, then the angle opposite the longer side
larger than the angle opposite the shorter
side.
Prove and apply the Pythagorean theorem
and its converse
Triangle Exterior Angle conjecture: The
measure of an exterior angle of a triangle is
equal to the sum of the measures of the
remote interior angles.
Pythagorean theorem:
Converse of the Pythagorean theorem:
If the lengths of the three sides of a triangle
satisfy the Pythagorean theorem equation
then the triangle is a right triangle.
 If a2 + b2 > c2 then the triangle is
acute
 If a2 + b2 < c2 then the triangle is
obtuse
State and apply the relationships that exist
when the altitude is drawn to the
hypotenuse of a right triangle
Use special right triangles (30˚-60˚-90˚ and
45˚-45˚-45˚) to solve problems.
Define and identify: circumference, radius,
diameter, arc, arc length, chord, secant,
tangent, and concentric circles.
In a right triangle, the altitude drawn to the
hypotenuse is the geometric mean of the
two pieces of the hypotenuse
 Geometry mean: when you multiply
two numbers and find the square root
of their product.
 Mean/Average: when you add two or
more numbers and divide by the total
numbers in the set.
30˚-60˚-90˚
 The short leg is _____ the hypotenuse
 The long leg is ________
 The hypotenuse is _____ the short leg
45˚-45˚-90˚
 The hypotenuse is the short leg
multiplied by ______.
Circumference:
The perimeter of a circle defined by the
equation __________ or _________
Radius:
 The distance from the center of a
circle to a point on the circle.
 The radius in of a circle does not
change no matter where you move
the point on the circle.
 The radius is ________ the diameter
Arc:
Two points on a circle and the continuous
part of the circle between them.
Arc length:
The portion of the circumference of the
circle described by an arc, measured in
units of length.
 the formula for arc length is:
 The formula for a sector is:
Chord:
A line segment whose endpoints lie on the
circle
Secant:
Determine and use measures of arcs and
related angles (central, inscribed, and
intersections of secants and tangents)
A line the intersects the circle in two points
Tangent:
A line that touches a point on a circle only
once.
 a tangent to a circle is __________________
to the radius drawn to the point of
tangency
 Tangent segments to a circle from a
point outside the circle are
_____________________.
Concentric Circles:
Circles that share the same center
 The area between concentric circles is
called a/n _________________
 To find the area of the annulus you
need to subtract the area of the
__________ circle from the area of the
__________ circle.
 Central angles are _______________ the
measure of the intercepted arc
 Inscribed angles are _____________ the
measure of the intercepted arc.
 List all conjectures and equations for
intersection of secants, chords, and
tangents: (chapter 6 pg. 310)
Given the center and the radius, find the
equation of a circle in the coordinate plane,
or given the equation of a circle in center
radius form, state the center and the radius
of the circle (chapter 5 pg. 504)

Describe the relationships between the
faces, edges, and vertices of polyhedral
(chapter 10)

Explain and use formulas for lateral area,
surface area, and volume of solids.
List all formulas for surface area:
List all formulas for volume:
Determine how changes in dimensions
affect the surface area and volume of
common geometric solids
Define and use trigonometric ratios (sine,
cosine, tangent, cotangent, secant, and
cosecant) in term of right triangles.