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Transcript
```Vocabulary
Construct
Congruent
Angle
Arc
Circle
Point
Compass
Distance
Perpendicular
Parallel
Transversal
Triangle
Equilateral
Equilateral
triangle
Point of
concurrency
Median
Altitude
Perpendicular
bisector
Line segment
Concepts/Topics
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NY State Performance Indicators
Skills to Review
Project/Lab
UNIT 1: CONSTRUCTION 9/13 –
CHAPTER 1
G.G.17
Construct a bisector of a given
Compasses help us create congruent arcs or
angle, using a straightedge and compass,
circles that intersect at points that allow us
and justify the construction
to bisect angles.
G.G.19 Construct lines parallel (or
A compass allows us to replicate the
perpendicular) to a given line through a
distance of two points
given point, using a straightedge and
Using arcs or circles greater than ½ the
given distance helps identify perpendicular compass, and justify the construction
G.G.18 Construct the perpendicular
lines
bisector of a given segment, using a
If 2 lines cut by a transversal make
congruent angles, then the lines are parallel straightedge and compass, and justify the
construction
(the goal of the construction of a parallel
G.G.20 Construct an equilateral triangle,
line is to construct a pair of congruent
using a straightedge and compass, and
angles)
justify the construction
Equilateral triangles have equal sides.
G.G.21 Investigate and apply the
These sides will be radii of congruent
concurrence of medians, altitudes, angle
circles that can be constructed using the
bisectors, and perpendicular bisectors of
compass.
triangles
Triangles can have up to 4 points of
concurrency
Perpendicular bisectors of a triangle share
the properties of both medians and
altitudes
UNIT 2: REASONING AND PROOF
CHAPTER 2
G.G.24 Determine the negation of a
 Proof is a justification that is logically valid
statement and establish its truth value
and based on definitions, postulates, and
G.G.25 Know and apply the conditions
theorems.
under which a compound statement
 Logical arguments consist of a set of
(conjunction, disjunction, conditional,
premises or hypotheses and a conclusion.
1
Parallel
Perpendicular
Slope
Line
Line segment
Point
Equation
Midpoint
Bisect
Bisector
Length
Distance
Endpoint
Perpendicular
bisector
Distance formula
Midpoint formula
biconditional) is true
 Reasoning and proofs are fundamental to
mathematics and help us prove or disprove G.G.26 Identify and write the inverse,
converse, and contrapositive of a given
various conjectures.
conditional statement and note the
 Only 1 counterexample is necessary to
prove a statement false, but 1 example does logical equivalences
G.G.27 Write a proof arguing from a given
not prove the truth value of a statement.
hypothesis to a given conclusion
 Inverse, converse, and contrapositive
statements relate to the original conditional
statement and vary in whether they are
always logically valid.
UNIT 3: PARALLEL AND PERPENDICULAR LINES
CHAPTER 3
Introduce formal proofs here
G.G.35 Determine if two lines cut by a
transversal are parallel, based on the
 Lines are parallel if alternate interior and
measure of given pairs of angles formed
exterior angles are congruent
by the transversal and the lines
 Alternate interior and exterior angles are
G.G.62 Find the slope of a perpendicular
congruent if lines are parallel (this is the
line, given the equation of a line
converse of statement 1)
G.G.63 Determine whether two lines are
 There is a special relationship between the
parallel, perpendicular, or neither, given
slopes of 2 parallel lines and the slopes of 2
their equations
perpendicular lines.
G.G.64 Find the equation of a line, given
 Slopes of parallel lines are equal
a point on the line and the equation of a
 Slopes of perpendicular lines are opposite
line perpendicular to the given line
reciprocals of one another
 We can use the relationship of the slopes of G.G.65 Find the equation of a line, given
a point on the line and the equation of a
parallel and perpendicular lines to prove
line parallel to the desired line
whether two equations produce parallel or
G.G.66 Find the midpoint of a line
perpendicular lines
segment, given its endpoints
 The relationship of the slopes of parallel
and perpendicular lines allows one to graph G.G.67 Find the length of a line segment,
given its endpoints
and find the equations of lines
G.G.68 Find the equation of a line that is
 The mid-points formula finds the halfway
the perpendicular bisector of a line
point for the x- and y-coordinates
segment, given the endpoints of the line
 The distance formula is derived from the
segment
Pythagorean theorem (the distance is the
hypotenuse of the right triangle formed by
Write a proof arguing from a given hypo
Slope
Finding the
equation of a line
given two points
Finding the
equation of a line
given slope and
one point
Pythagorean
Theorem
2
the x- and y-coordinates)
 We can use the midpoint and our
understanding of perpendicular lines to
find the equation of a perpendicular
bisector within a coordinate plane
UNIT 4: TRIANGLE PROPERTIES AND CONGRUENCE
CHAPTERS 4 & 5
What does it mean for triangles to be
congruent?
G.28 Determine the congruence of two
triangles using one of the five congruence
techniques (SSS, SAS, ASA, AAS, HL) given
How can I prove that two triangles are
congruent?
and/or angles of two congruent triangles
G.29 Identify corresponding parts of
What are some of the defining characteristics
congruent triangles
of a triangle?
G.30 Investigate, justify, and apply
theorems about the sum of the measures
What characterizes different types of
of the angles of a triangle
triangles?
G.32 Investigate, justify, and apply
using the exterior angle theorem
G.31 Investigate, justify and apply the
isosceles triangle theorem and its
converse
G.33 Investigate, justify and apply the
triangle inequality theorem
G.34 Determine the longest side of a
triangle given the three angle measures
or the largest angle given the lengths of
three sides of a triangle
UNIT 5: TRIANGLE SIMILARITY
CHAPTER 8
3
Alternate interior
angles
Alternate exterior
angles
Corresponding
angles
Vertical angles
Supplementary
angles
Linear pairs
Parallelogram
Diagonal
Opposite angles
Rhombus
Square
G.44 Establish similarity of triangles,
using the following theorems: AA, SAS,
and SSS
G.45 Investigate, justify, and apply
G.46 Investigate, justify, and apply
relationships among the segments of the
sides of the triangle, given one or more
lines parallel to one side of a triangle and
intersecting the other two sides of the
triangle
G.42 Investigate, justify, and apply
based on the properties of the line
segment joining the midpoints of two
sides of the triangle
G.47 Investigate, justify, and apply
CHAPTER 6
G.G.38 Investigate, justify, and apply
 Parallelograms:
have 2 sets of parallel sides
their angles, sides, and diagonals
diagonals bisect one another
G.G.39 Investigate, justify, and apply
have opposite angles that are always
congruent
(rectangles, rhombuses, squares)
 Rhombuses and squares have congruent
involving their angles, sides, and
sides
diagonals
 Rectangles and squares have 4 right angles
G.G.40 Investigate, justify, and apply
includes more specific 4-sided polygons like theorems about trapezoids (including
isosceles trapezoids) involving their
parallelograms, rhombuses, rectangles,
angles, sides, medians, and diagonals
squares, and trapezoids
 The sum of a polygons’ interior angles is
are parallelograms, rhombuses,
determined by 180(n-2)
rectangles, squares, or trapezoids
 A polygon’s exterior angles can have a
4
Rectangle
Right angle
Polygon
Trapezoid
Interior angle
Exterior angle
negative or a positive measurement as they
are always measured counterclockwise.
 Every pair of interior and exterior angles
are supplementary (add up to 180 degrees)





G.G.36 Investigate, justify, and apply
theorems about the sum of the measures
of the interior and exterior angles of
polygons
G.G.37 Investigate, justify, and apply
exterior angle measure of regular
polygons
UNIT 7: TRANSFORMATIONS
G.54 Define, investigate, justify, and apply
An isometry of the coordinate plane is a
isometries in the plane (rotations,
linear transformation which preserves
reflections, translations, glide reflections)
length. Put another way, isometries
Note: Use proper function notation.
produce congruent figures.
G.55 Investigate, justify, and apply the
We can use our existing knowledge of
properties that remain invariant under
perpendicularity, parallelism, and
translations, rotations, reflections, and
glide reflections
the results of various transformations.
G.56 Identify specific isometries by
Not all transformations are isometries.
observing orientation, numbers of
Dilations produce figures that are similar,
invariant points, and/or parallelism
but not congruent.
G.57 Justify geometric relationships
It is possible to perform more than one
transformation on the same figure to obtain (perpendicularity, parallelism,
congruence) using transformational
the image from the preimage
techniques (translations, rotations,
Reflections result in symmetry about a
reflections)
point or line.
G.58 Define, investigate, justify, and apply
similarities (dilations and the
composition of dilations and isometries)
G.59 Investigate, justify, and apply the
properties that remain invariant under
similarities
G.60 Identify specific similarities by
observing orientation, numbers of
invariant points, and/or parallelism
G.61 Investigate, justify, and apply the
analytical representations for
5






translations, rotations about the origin of
90º and 180º, reflections over the lines
x=0, y=0 , and y=x , and dilations
centered at the origin
UNIT 8: CIRCLES AND LOCI
CHAPTER 10
G.71 Write the equation of a circle, given
A circle includes all points equidistant, r,
its center and radius or given the
from the center (h, k).
endpoints of a diameter
G.72
The equation of a circle includes the center
Write the equation of a circle, given its
graph
We can figure out the location of the center
Note: The center is an ordered pair of
and the length of the radius of a circle
simply by looking at the equation of a circle integers and the radius is an integer.
The center (h, k) has the opposite sign from
G.73 Find the center and radius of a
how it appears in the equation
The use of square roots allows us to find the circle, given the equation of the circle in
center of the circle.
G.74 Graph circles of the form
Loci are sets of points that satisfy 1 given
condition, usually involving a distance from
another object. There are 5 main scenarios
that loci problems involve.
 Compound loci are sets of points that
satisfy more than 1 given condition.
Diameter
Chord
Center
Diameter
Tangent line
Arc length
Circumference
Pi
Arc
Secant
(x h)2 (y k)2 r2
G.22 Solve problems using compound
loci
G.23 Graph and solve compound loci in
the coordinate plane
UNIT 9: TANGENTS, SECANTS, AND CHORDS
G.G.49 Investigate, justify, and apply
 Diameters are the longest chords in any
theorems regarding chords of a circle:
circle.
O perpendicular bisectors of chords
 Chords are related uniquely to circles and
O the relative lengths of chords as
their distance from the center
compared to their distance from the
 Perpendicular bisectors of chords also
center of the circle
bisect the circle. Put another way, if a
G.G.50 Investigate, justify, and apply
diameter is perpendicular to a chord, it
theorems about tangent lines to a circle:
bisects that chord.
O a perpendicular to the tangent at the
 The radius is always perpendicular to the
point of tangency
tangent line
O two tangents to a circle from the same
 Every point outside a circle can have
6
Ray
exactly two tangent lines extended to the
circle
 Arc lengths are related to the entire circle
as a fraction of the overall circumference of
the circle (n/360 * 2∏r)
 Chords that are parallel create congruent
arcs.
 We can make generalizations about the
length of line segments based upon the
number of times they intersect a circle and
whether the intersection of 2 line segments
is inside or outside the circle.
external point
O common tangents of two nonintersecting or tangent circles
G.G.51 Investigate, justify, and apply
theorems about the arcs determined by
the rays of angles formed by two lines
intersecting a circle when the vertex is:
O inside the circle (two chords)
O on the circle (tangent and chord)
O outside the circle (two tangents, two
secants, or tangent and secant)
G.G.52 Investigate, justify, and apply
theorems about arcs of a circle cut by two
parallel lines
G.G.53 Investigate, justify, and apply
theorems regarding segments intersected
by a circle:
O along two tangents from the same
external point
O along two secants from the same
external point
O along a tangent and a secant from the
same external point
O along two intersecting chords of a
given circle
UNIT 10: COORDINATE PROOFS
 Wrap up year with coordinate proofs to
G.G.69 Investigate, justify, and apply
review coordinate geometry, parallel and
the properties of triangles and
perpendicular lines, and properties of
plane, using the distance, midpoint,
and slope formulas
UNIT 11: AREA, VOLUME, AND PERIMETER
CHAPTERS 11 & 12
7
3-Dimensional
2-Dimensional
Volume
Area
Perimeter
Prism
Base
Height
Altitude
Face
Regular pyramid
Cylinder
Right circular
cone
Sphere
Plane
Line
Point
Intersection
Perpendicular
Contain
Coplanar
Normal
G.G.10 Know and apply that the lateral
 Volume is a measure of 3-dimensional
edges of a prism are congruent and
space like the amount of liquid that a
parallel
container could hold.
G.G.11 Know and apply that two prisms
 Volume of a prism is the area of the base
have equal volumes if their bases have
times the height (or altitude) (v = l * w * h)
 A pyramid has a square or rectangular base equal areas and their altitudes are equal
G.G.12 Know and apply that the volume
and four slanted, triangular sides. At a
minimum, the opposing faces are congruent of a prism is the product of the area of the
base and the altitude
 Lateral area is the area of the lateral faces.
G.G.13 Apply the properties of a regular
Surface area is lateral area plus the area of
pyramid
the base.
G.G.14 Apply the properties of a
 “Height” refers to the perpendicular
cylinder
distance from the center of the base to the
common vertex where all lateral faces meet. G.G.15 Apply the properties of a right
circular cone
 “Slant height” refers to the distance along a
G.G.16 Apply the properties of a sphere
lateral face from the base to the common
vertex where all lateral faces meet.
 Volume of a cylinder is the area of the base
times the height (or altitude) (v = ∏r2 * h)
 A cone is made of two surface pieces—a
circular base and an additional lateral
surface
 The cross section of a sphere is a circle
 A sphere consists of all the points whose
distance from the center is less than or
UNIT 12: LINES AND PLANES
 Points, lines, segments, and planes are basic G.G.1 Know and apply that if a line is
perpendicular to each of two intersecting
geometric structures and allow us to make
lines at their point of intersection, then
conclusions about more complex 2- and 3the line is perpendicular to the plane
dimensional objects
determined by them
 A plane can be defined by 3 points, 2
G.G.2 Know and apply that through a
intersecting lines or a line and point not on
given point there passes one and only
the line
one plane perpendicular to a given line
 A line is defined by 2 points
G.G.3 Know and apply that through a
 Coplanar means that the given lines or
8
points lie within the same plane.
 When a plane contains a given line, all lines
and planes that are perpendicular to the
given line are also perpendicular to the
plane
 If a line is normal to one of two parallel
plane, and that line does not lie in either
plane, then that line is normal to the other
of the two parallel planes
given point there passes one and only
one line perpendicular to a given plane
G.G.4 Know and apply that two lines
perpendicular to the same plane are
coplanar
G.G.5 Know and apply that two planes
are perpendicular to each other if and
only if one plane contains a line
perpendicular to the second plane
G.G.6 Know and apply that if a line is
perpendicular to a plane, then any line
perpendicular to the given line at its
point of intersection with the given plane
is in the given plane
G.G.7 Know and apply that if a line is
perpendicular to a plane, then every
plane containing the line is perpendicular
to the given plane
G.G.8 Know and apply that if a plane
intersects two parallel planes, then the
intersection is two parallel lines
G.G.9 Know and apply that if two
planes are perpendicular to the same
line, they are parallel
UNIT 13: SYSTEMS OF EQUATIONS
G.G.70 Solve systems of equations
involving one linear equation and one