Download Use coordinates to prove simple geometric theorems algebraically

Mathematics Standards Comparison
GEOMETRY
Next Generation Sunshine State
Standards
Body of Knowledge: Geometry
Standard 1: Points, Lines, Angles,
and Planes
BENCHMARK
CODE
MA.912.G.1.1
Common Core State Standards
Conceptual Categories: Geometry
BENCHMARK
DOMAIN
CLUSTER
STANDARD
Find the lengths and
midpoints of line segments
in two-dimensional
coordinate systems.
G-GPE
Expressing
Geometric
Properties with
Equations
Use coordinates to
prove simple geometric
theorems algebraically
7. Use coordinates to compute perimeters of polygons and areas of triangles
and rectangles, e.g., using the distance formula.
Use coordinates to
prove simple geometric
theorems algebraically
MA.912.G.1.2
Not Assessed
MA.912.G.1.3
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
Construct congruent
segments and angles,
angle bisectors, and
parallel and perpendicular
lines using a straightedge
and compass or a drawing
program, explaining and
justifying the process used.
G-CO
Congruence
Make geometric
constructions
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
Identify and use the
relationships between
special pairs of angles
formed by parallel lines and
transversals.
G-CO
Congruence
Prove geometric
theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles
are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints
*In 8th grade: 8.G.5 Use informal arguments to establish facts about the angles created
when parallel lines are cut by a transversal
Body of Knowledge: Geometry
Standard 2: Polygons
BENCHMARK
CODE
MA.912.G.2.1
Assessed with
MA.912.G.2.3.
BENCHMARK
Identify and describe
convex, concave, regular,
and irregular polygons.
Conceptual Categories: Functions
DOMAIN
CLUSTER
STANDARD
There is no Benchmark – Common Core Alignment
MA.912.G.2.2
Determine the measures of
interior and exterior angles
of polygons, justifying the
method used.
G-CO
Congruence
Prove geometric
theorems
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point
*In 8th grade: 8.G.5 5. Use informal arguments to establish facts about the angle sum
and exterior angle of triangles.
MA.912.G.2.3
Also assesses
MA.912.G.2.1,
MA.912.G.4.1/4.2
MA.912.G.4.4/4.5
MA.912.G.2.4
Use properties of congruent
and similar polygons to
solve mathematical or realworld problems.
Apply transformations
(translations, reflections,
rotations, dilations, and
scale factors) to polygons
to determine congruence,
similarity, and symmetry.
Know that images formed
by translations, reflections,
and rotations are congruent
to the original shape.
Create and verify
tessellations of the plane
using polygons.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
G-CO
Congruence
Understand similarity in
terms of similarity
transformations
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs
of sides
Prove theorems
involving similarity
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures
Experiment with
transformations in the
plane.
2. Represent transformations in the plane using, e.g., transparencies and
geometry software; describe transformations as functions that take points in the
plane as inputs and give other points as outputs.
Compare transformations that preserve distance and angle to those that do not
(e.g., translation versus horizontal stretch).
3. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the
rotations and reflections that carry it onto itself.
4. Develop definitions of rotations, reflections, and translations in terms of
angles, circles, perpendicular lines, parallel lines, and line segments.
5. Given a geometric figure and a rotation, reflection, or translation, draw the
transformed figure using, e.g., graph paper, tracing paper, or geometry
software. Specify a sequence of transformations that will carry a given figure
onto another.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Understand congruence
in terms of rigid
motions
6. Use geometric descriptions of rigid motions to transform figures and to predict
the effect of a given rigid motion on a given figure; given two figures, use the
definition of congruence in terms of rigid motions to decide if they are congruent.
Understand similarity in
terms of similarity
transformations
1. Verify experimentally the properties of dilations given by a center and a scale
factor:
a.
b.
A dilation takes a line not passing through the center of the dilation to a
parallel line, and leaves a line passing through the center unchanged.
The dilation of a line segment is longer or shorter in the ratio given by
the scale factor
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs
of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
MA.912.G.2.5
Also assesses
MA.912.G.2.7
MA.912.G.2.6
Honors
MA.912.G.2.7
Assessed with
MA.912.G.2.5,
MA.912.G.7.7
Explain the derivation and
apply formulas for
perimeter and area of
polygons (triangles,
quadrilaterals, pentagons,
etc.).
Use coordinate geometry to
prove properties of
congruent, regular and
similar polygons, and to
perform transformations in
the plane.
G-GPE
Expressing
Geometric
Properties with
Equations
Use coordinates to
prove simple geometric
theorems algebraically
7. Use coordinates to compute perimeters of polygons and areas of triangles
and rectangles, e.g., using the distance formula.
G-GPE
Expressing
Geometric
Properties with
Equations
Use coordinates to
prove simple geometric
theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on
the circle centered at the origin and containing the point (0, 2).
*In 7th Grade: 7.G.1 Solve problems involving scale drawings of geometric figures,
Determine how changes in
dimensions affect the
perimeter and area of
common geometric figures.
including computing actual lengths and areas from a scale drawing and reproducing a
scale drawing at a different scale.
Body of Knowledge: Geometry
Standard 3: Quadrilaterals
BENCHMARK
CODE
MA.912.G.3.1
Assessed with
MA.912.G.3.4.
MA.912.G.3.2
Assessed with
MA.912.G.3.4.
MA.912.G.3.4
Also assesses
MA.912.D.6.4,
MA.912.G.8.5.
Conceptual Categories: Algebra/Functions
BENCHMARK
DOMAIN
CLUSTER
STANDARD
Describe, classify, and
compare relationships
among quadrilaterals
including the square,
rectangle, rhombus,
parallelogram, trapezoid,
and kite.
G-CO
Congruence
Prove geometric
theorems
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with congruent
diagonals
Compare and contrast
special quadrilaterals on
the basis of their properties.
Prove theorems involving
quadrilaterals.
MA.912.G.3.3
Use coordinate geometry to
prove properties of
congruent, regular, and
similar quadrilaterals.
G-GPE
Expressing
Geometric
Properties with
Equations
Body of Knowledge: Geometry
Standard 4: Triangles
BENCHMARK
CODE
MA.912.G.4.1
Assessed with
MA.912.G.2.3.
BENCHMARK
Use coordinates to
prove simple geometric
theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on
the circle centered at the origin and containing the point (0, 2).
Conceptual Categories: Algebra/ Functions/Geometry/Statistics
DOMAIN
CLUSTER
STANDARD
*Classifying triangles is done in 4th grade: 4.G.2 Classify two-dimensional
figures based on the presence or absence of parallel or perpendicular lines, or
the presence or absence of angles of a specified size. Recognize right triangles
as a category, and identify right triangles
Classify, construct, and
describe triangles that are
right, acute, obtuse,
scalene, isosceles,
equilateral, and
equiangular.
*Did not find mention of classification based on side lengths: scalene,
isosceles, equilateral.
*Constructing triangles done in 7th grade: 7.G.2 Draw (freehand, with ruler and
protractor, and with technology) geometric shapes with given conditions. Focus
on constructing triangles from three measures of angles or sides, noticing when
the conditions determine a unique triangle, more than one triangle, or no
triangle.
MA.912.G.4.2
Assessed with
MA.912.G.2.3.
Define, identify, and
construct altitudes,
medians, angle bisectors,
perpendicular bisectors,
orthocenter, centroid,
incenter, and circumcenter.
G-CO
Congruence
Prove geometric
theorems
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point.
Make geometric
constructions
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
*Did not find mention of orthocenter, centroid, incenter, and circumcenter in
CCSS.
MA.912.G.4.3
Not assessed.
MA.912.G.4.4
Assessed with
MA.912.G.2.3.
There is no Benchmark – Common Core Alignment
Construct triangles
congruent to given
triangles.
Use properties of congruent
and similar triangles to
solve problems involving
lengths and areas.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Understand similarity in
terms of similarity
transformations
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs
of sides
MA.912.G.4.5
Assessed with
MA.912.G.2.3.
MA.912.G.4.6
Also assesses
MA.912.D.6.4 and
MA.912.G.8.5.
Apply theorems involving
segments divided
proportionally.
Prove that triangles are
congruent or similar and
use the concept of
corresponding parts of
congruent triangles.
Prove theorems
involving similarity
5. Use congruence and similarity criteria for triangles to solve problems and to
prove relationships in geometric figures
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Prove theorems
involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity
G-GPE
Expressing
Geometric
Properties with
Equations
Use coordinates to
prove simple geometric
theorems algebraically
6. Find the point on a directed line segment between two given points that
partitions the segment in a given ratio.
G-CO
Congruence
Understand congruence
in terms of rigid
motions
7. Use the definition of congruence in terms of rigid motions to show that two
triangles are congruent if and only if corresponding pairs of sides and
corresponding pairs of angles are congruent.
8. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow
from the definition of congruence in terms of rigid motions.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Understand similarity in
terms of similarity
transformations
2. Given two figures, use the definition of similarity in terms of similarity
transformations to decide if they are similar; explain using similarity
transformations the meaning of similarity for triangles as the equality of all
corresponding pairs of angles and the proportionality of all corresponding pairs
of sides.
3. Use the properties of similarity transformations to establish the AA criterion
for two triangles to be similar.
MA.912.G.4.7
Apply the inequality
theorems: triangle
inequality, inequality in one
triangle, and the Hinge
Theorem.
MA.912.G.4.8
Honors
Use coordinate geometry to
prove properties of
congruent, regular, and
similar triangles.
There is no Benchmark – Common Core Alignment
G-GPE
Expressing
Geometric
Properties with
Equations
Body of Knowledge: Geometry
Standard 5: Right Triangles
BENCHMARK
CODE
MA.912.G.5.1
Assessed with
Use coordinates to
prove simple geometric
theorems algebraically
4. Use coordinates to prove simple geometric theorems algebraically. For
example, prove or disprove that a figure defined by four given points in the
coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on
the circle centered at the origin and containing the point (0, 2).
Conceptual Categories: Algebra/Functions
BENCHMARK
DOMAIN
CLUSTER
STANDARD
Prove and apply the
Pythagorean Theorem and
its converse.
G-SRT
Similarity, Right
Triangles, &
Prove theorems
involving similarity
4. Prove theorems about triangles. Theorems include: a line parallel to one side
of a triangle divides the other two proportionally, and conversely; the
Pythagorean Theorem proved using triangle similarity.
Trigonometry
MA.912.G.5.4.
Define trigonometric
ratios and solve
problems involving right
triangles
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems
*Students also work with the Pythagorean Theorem in 8 th Grade.
MA.912.G.5.2
Assessed with
MA.912.G.5.4.
MA.912.G.5.3
Assessed with
MA.912.G.5.4.
There is no Benchmark – Common Core Alignment
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° - 90°) to solve
problems.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Define trigonometric
ratios and solve
problems involving right
triangles
6. Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute
angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles
in applied problems.
MA.912.G.5.4
Solve real-world problems
involving right triangles.
*Found in 4th year HS math course: F-TF Extend the domain of
trigonometric functions using the unit circle.
Also assesses
MA.912.G.5.1,
MA.912.G.5.2,
MA.912.G.5.3.
3. (+) Use special triangles to determine geometrically the values of sine,
cosine, tangent for π/3, π/4 and π/6, and use the unit circle to express the
values of sine, cosines, and tangent for x, π + x, and 2π – × in terms of their
values for x, where x is any real number. (+) = Honors
Body of Knowledge: Geometry
Standard 6: Circles
BENCHMARK
CODE
MA.912.G.6.1
Conceptual Categories: Number and Quantity/Algebra
BENCHMARK
DOMAIN
CLUSTER
STANDARD
Determine the center of a
given circle. Given three
points not on a line,
construct the circle that
passes through them.
Construct tangents to
circles. Circumscribe and
inscribe circles about and
within triangles and regular
polygons.
G-C
Circles
Understand and apply
theorems about circles
3. Construct the inscribed and circumscribed circles of a triangle, and prove
properties of angles for a quadrilateral inscribed in a circle.
4. (+) Construct a tangent line from a point outside a given circle to the circle.
(+) = Honors
G-CO
Congruence
Make geometric
constructions
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle
MA.912.G.6.2
Assessed with
MA.912.G.6.5.
MA.912.G.6.4
Assessed with
MA.912.G.6.5.
MA.912.G.6.5
Define and identify:
circumference, radius,
diameter, arc, arc length,
chord, secant, tangent and
concentric circles.
G-C
Circles
Determine and use
measures of arcs and
related angles (central,
inscribed, and intersections
of secants and tangents).
Understand and apply
theorems about circles
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
Find arc lengths and
areas of sectors of
circles
5. Derive using similarity the fact that the length of the arc intercepted by an
angle is proportional to the radius, and define the radian measure of the angle
as the constant of proportionality; derive the formula for the area of a sector.
Solve real-world problems
using measures of
circumference, arc length,
and areas of circles and
sectors.
MA.912.G.6.3
Honors
Prove theorems related to
circles, including related
angles, chords, tangents,
and secants.
G-C
Circles
Understand and apply
theorems about circles
2. Identify and describe relationships among inscribed angles, radii, and chords.
Include the relationship between central, inscribed, and circumscribed angles;
inscribed angles on a diameter are right angles; the radius of a circle is
perpendicular to the tangent where the radius intersects the circle.
MA.912.G.6.6
Also assesses
MA.912.G.6.7.
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.
G-GPE
Expressing
Geometric
Properties with
Equations
Translate between the
geometric description
and the equation for a
conic section
1. Derive the equation of a circle of given center and radius using the
Pythagorean Theorem; complete the square to find the center and radius of a
circle given by an equation.
MA.912.G.6.7
Given the equation of a
circle in center-radius form
or given the center and the
radius of a circle, sketch
the graph of the circle.
Found in Algebra I: A-REI Solve systems of equations.
7. Solve a simple system consisting of a linear equation and a quadratic
equation in two variables algebraically and graphically. For example, find the
points of intersection between the line y = –3x and the circle x2 + y2 = 3
Body of Knowledge: Geometry
Standard 7: Polyhedra and Other
Solids
BENCHMARK
CODE
BENCHMARK
Conceptual Categories: Algebra
DOMAIN
CLUSTER
STANDARD
*In 6th grade: Solve real-world and mathematical problems involving area,
surface area, and volume.
4. Represent three-dimensional figures using nets made up of rectangles and
triangles, and use the nets to find the surface area of these figures. Apply these
techniques in the context of solving real-world and mathematical problems.
MA.912.G.7.1
Also assesses
MA.912.G.7.2.
Describe and make regular,
non-regular, and oblique
polyhedra, and sketch the
net for a given polyhedron
and vice versa.
MA.912.G.7.2
Describe the relationships
between the faces, edges,
and vertices of polyhedra.
MA.912.G.7.3
Honors
Identify, sketch, find areas
and/or perimeters of cross
sections of solid objects.
G-GMD
Geometric
Measurement &
Dimension
Visualize relationships
between twodimensional and threedimensional objects
4. Identify the shapes of two-dimensional cross-sections of three-dimensional
objects, and identify three-dimensional objects generated by rotations of twodimensional objects
MA.912.G.7.4
Assessed with
MA.912.G.7.5.
Identify chords, tangents,
radii, and great circles of
spheres.
G-GMD
Geometric
Measurement &
Dimension
Explain volume
formulas and use them
to solve problems
1. Give an informal argument for the formulas for the circumference of a circle,
area of a circle, volume of a cylinder, pyramid, and cone. Use dissection
arguments, Cavalieri’s principle, and informal limit arguments.
MA.912.G.7.5
Explain and use formulas
for lateral area, surface
area, and volume of solids.
MA.912.G.7.6
Assessed with
MA.912.G.7.5.
Identify and use properties
of congruent and similar
solids.
MA.912.G.7.7
Determine how changes in
dimensions affect the
surface area and volume of
common geometric solids.
Also assesses
MA.912.G.2.7.
2. (+) Give an informal argument using Cavalieri’s principle for the formulas for the volume
of a sphere and other solid figures.
3. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.
There is no Benchmark – Common Core Alignment
Body of Knowledge: Algebra
Standard 8: Mathematical Reasoning
and Problem Solving
BENCHMARK
CODE
MA.912.G.8.1
Embedded
throughout.
BENCHMARK
Analyze the structure of
Euclidean geometry as an
axiomatic system.
Distinguish between
undefined terms,
definitions, postulates, and
theorems.
Conceptual Categories: Number and Quantity
DOMAIN
CLUSTER
STANDARD
MA.912.G.8.2
Embedded
throughout.
MA.912.G.8.3
Embedded
throughout.
MA.912.G.8.4
MA.912.G.8.5
Assessed with
MA.912.G.3.4
MA.912.G.4.6.
MA.912.G.8.6
Not assessed.
1 Make sense of problems and persevere in solving them.
Use a variety of problem
solving strategies, such as
drawing a diagram, making
a chart, guess-and-check,
solving a simpler problem,
writing an equation, and
working backwards.
Mathematically proficient students start by explaining to themselves the meaning of a
problem and looking for entry points to its solution. They analyze givens, constraints,
relationships, and goals. They make conjectures about the form and meaning of the
solution and plan a solution pathway rather than simply jumping into a solution attempt.
They consider analogous problems, and try special cases and simpler forms of the original
problem in order to gain insight into its solution. They monitor and evaluate their progress
and change course if necessary. Older students might, depending on the context of the
problem, transform algebraic expressions or change the viewing window on their graphing
calculator to get the information they need. Mathematically proficient students can explain
correspondences between equations, verbal descriptions, tables, and graphs or draw
diagrams of important features and relationships, graph data, and search for regularity or
trends. Younger students might rely on using concrete objects or pictures to help
conceptualize and solve a problem. Mathematically proficient students check their
answers to problems using a different method, and they continually ask themselves,
“Does this make sense?” They can understand the approaches of others to solving
complex problems and identify correspondences between different approaches.
Determine whether a
solution is reasonable in
the context of the original
situation.
Make conjectures with
justifications about
geometric ideas.
Distinguish between
information that supports a
conjecture and the proof of
a conjecture.
G-CO
Congruence
Prove geometric
theorems
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point.
Write geometric proofs,
including proofs by
contradiction and proofs
involving coordinate
geometry. Use and
compare a variety of ways
to present deductive proofs,
such as flow charts,
paragraphs, two column,
and indirect proofs.
Perform basic constructions
using straightedge and
compass, and/or drawing
programs describing and
justifying the procedures
used. Distinguish between
sketching, constructing,
and drawing geometric
figures.
9. Prove theorems about lines and angles. Theorems include: vertical angles
are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with congruent
diagonals
G-CO
Congruence
Make geometric
constructions
12. Make formal geometric constructions with a variety of tools and methods
(compass and straightedge, string, reflective devices, paper folding, dynamic
geometric software, etc.). Copying a segment; copying an angle; bisecting a
segment; bisecting an angle; constructing perpendicular lines, including the
perpendicular bisector of a line segment; and constructing a line parallel to a
given line through a point not on the line.
13. Construct an equilateral triangle, a square, and a regular hexagon inscribed
in a circle
Body of Knowledge: Trigonometry
Standard 2: Trigonometry in
Triangles
BENCHMARK
CODE
MA.912.T.2.1
Conceptual Categories: Functions/ Algebra
BENCHMARK
DOMAIN
CLUSTER
STANDARD
Define and use the
trigonometric ratios (sine,
cosine, tangent, cotangent,
secant, cosecant) in terms
of angles of right triangles.
G-SRT
Similarity, Right
Triangles, &
Trigonometry
Define trigonometric
ratios and solve
problems involving right
triangles
6. Understand that by similarity, side ratios in right triangles are properties of the
angles in the triangle, leading to definitions of trigonometric ratios for acute
angles.
7. Explain and use the relationship between the sine and cosine of
complementary angles.
8. Use trigonometric ratios and the Pythagorean Theorem to solve right
triangles in applied problems.
Mathematical Practice Standard 1
Body of Knowledge: Discrete
Standard 6: Logic
BENCHMARK
CODE
MA.912.D.6.2
Also assesses
MA.912.D.6.3.
BENCHMARK
MA.912.D.6.3
Determine whether two
propositions are logically
equivalent.
MA.912.D.6.4
Use methods of direct and
indirect proof and
determine whether a short
proof is logically valid.
Assessed with
MA.912.G.3.4
MA.912.G.4.6.
DOMAIN
CLUSTER
STANDARD
There is no Benchmark – Common Core Alignment
Find the converse, inverse,
and contrapositive of a
statement.
G-CO
Congruence
Prove geometric
theorems
9. Prove theorems about lines and angles. Theorems include: vertical angles
are congruent; when a transversal crosses parallel lines, alternate interior
angles are congruent and corresponding angles are congruent; points on a
perpendicular bisector of a line segment are exactly those equidistant from the
segment’s endpoints
10. Prove theorems about triangles. Theorems include: measures of interior
angles of a triangle sum to 180°; base angles of isosceles triangles are
congruent; the segment joining midpoints of two sides of a triangle is parallel to
the third side and half the length; the medians of a triangle meet at a point.
11. Prove theorems about parallelograms. Theorems include: opposite sides are
congruent, opposite angles are congruent, the diagonals of a parallelogram
bisect each other, and conversely, rectangles are parallelograms with congruent
diagonals