# Geometry

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```					          INDIANA

DRAFT – MATHEMATICS – DRAFT

LIST OF TERMS AND CONCEPTS
FOR GEOMETRY

BY MCREL

NOVEMBER 2001
Summary of Standards for Geometry

Standard 1
Points, Lines, Angles, and Planes
Students find lengths and midpoints of lines. They describe and use parallel and perpendicular lines.
They find slopes and equations of lines.

Standard 2
Polygons
Students identify and describe polygons, and measure interior and exterior angles. They use congruence,
similarity, symmetry, tessellations, and transformations. They find measures of sides, perimeters, and
areas.

Standard 3
Students identify and describe simple quadrilaterals. They use congruence and similarity. They find
measures of sides, perimeters, and areas.

Standard 4
Triangles
Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle
bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They
apply inequality theorems.

Standard 5
Right Triangles
Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the
trigonometric relations sine, cosine, and tangent.

Standard 6
Circles
Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths, and
areas. They prove theorems about circles. They find equations of circles.

Standard 7
Polyhedra and Other Solids
Students describe and make polyhedra and other solids. They describe relationships and symmetries, and
use congruence and similarity.

Standard 8
Mathematical Reasoning and Problem Solving
Students use a variety of strategies to solve problems.
Students develop and evaluate mathematical arguments and proofs.
Standard 1
Points, Lines, Angles, and Planes

Students find lengths and midpoints of lines. They describe and use parallel and perpendicular lines.
They find slopes and equations of lines.

Find the lengths and midpoints of line segments in one- or two-dimensional coordinate systems.

    length of a line segment
    midpoint of a line segment
    one-dimensional coordinate system
    two-dimensional coordinate system

Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a
straight edge and compass, explaining and justifying the process used.

    angle [d]
    angle bisector [d]
    parallel lines [d]
    perpendicular lines [d]
    congruency [d]
    congruent segments
    congruent angles
    straight edge
    compass

Understand and use the relationships between special pairs of angles formed by parallel lines and
transversals.

    transversal
    angle [d]
    parallel lines [d]

Use coordinate geometry to find slopes, parallel lines, perpendicular lines, and equations of lines.

    coordinate geometry [d]
    parallel lines [d]
    perpendicular lines [d]
    slope [d]
    equation of a line [d]

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 1
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Standard 2
Polygons

Students identify and describe polygons, and measure interior and exterior angles. They use
congruence, similarity, symmetry, tessellations, and transformations. They find measures of sides,
perimeters, and areas.

Identify and describe convex, concave, and regular polygons.

    convex polygon
    concave polygon
    regular polygon

Find measures of interior and exterior angles of polygons, justifying the method used.

    interior angle
    exterior angle

Use properties of congruent and similar polygons to solve problems.

    congruency [d]
    similarity [d]

Apply transformations (slides, flips, turns, expansions, and contractions) to polygons in order to
determine congruence, similarity, symmetry, and tessellations. Know that images formed by slides, flips
and turns are congruent to the original shape.

    transformation
    slide (translation)
    flip (reflection)
    turn (rotation)
    similarity [d]
    symmetry
    expansion/contraction (dilation)
    tessellation
    congruency [d]

Find and use measures of sides, perimeters, and areas of polygons, and relate these measures to each
other using formulas.

    perimeter
    area [d]
    side of a polygon
    formula for perimeter of a polygon

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 2
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

    formula for area of a polygon
    perimeter of a polygon
    area of a polygon

Use coordinate geometry to prove properties of polygons such as regularity, congruence, and similarity.

    coordinate geometry [d]
    congruency [d]
    similarity [d]

Standard 3

Students identify and describe simple quadrilaterals. They use congruence and similarity. They find
measures of sides, perimeters, and areas.

Describe, classify, and understand relationships among the quadrilaterals square, rectangle, rhombus,
parallelogram, trapezoid, and kite.

    kite
    square
    rectangle
    rhombus
    parallelogram
    trapezoid

Use properties of congruent and similar quadrilaterals to solve problems involving lengths and areas.

    length
    area [d]

Find and use measures of sides, perimeters, and areas of quadrilaterals, and relate these measures to each
other using formulas.

    perimeter [d]
    area [d]
    formula for perimeter of a quadrilateral
    formula for area of a quadrilateral

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 3
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Use coordinate geometry to prove properties of quadrilaterals such as regularity, congruence, and
similarity.

    coordinate geometry [d]
    congruency [d]
    similarity [d]

Standard 4
Triangles

Students identify and describe types of triangles. They identify and draw altitudes, medians, and angle
bisectors. They use congruence and similarity. They find measures of sides, perimeters, and areas. They
apply inequality theorems.

Identify and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and
equiangular.

    triangle
    right triangle [d]
    acute triangle
    obtuse triangle
    scalene triangle
    isosceles triangle
    equilateral triangle
    equiangular triangle

Define, identify, and construct altitudes, medians, angle bisectors, and perpendicular bisectors.

    altitude [d]
    angle bisector [d]
    perpendicular bisector
    median of a shape

Construct triangles congruent to given triangles.

    triangle [d]
    congruent triangles [d]

Use properties of congruent and similar triangles to solve problems involving lengths and areas.

    triangle [d]
    congruent triangles [d]
    similar triangles

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 4
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Prove and apply theorems involving segments divided proportionally.

    theorem [d]
    proportional segment division

Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent
triangles.

    triangle [d]
    congruent triangles [d]
    congruency [d]

Find measures of sides, perimeters, and areas of triangles, and relate these measures to each other using
formulas.

    triangle [d]
    formula for perimeter of a triangle
    formula for area of a triangle

Prove, understand, and apply the inequality theorems: triangle inequality, inequality in one triangle, and
hinge theorem.

    triangle [d]
    triangle inequality theorem
    inequality in one triangle theorem
    hinge theorem

Use coordinate geometry to prove properties of triangles such as regularity, congruence, and similarity.

    triangle [d]
    coordinate geometry [d]
    congruency [d]
    similarity [d]

Standard 5
Right Triangles

Students prove the Pythagorean Theorem and use it to solve problems. They define and apply the
trigonometric relations sine, cosine, and tangent.

Prove and use the Pythagorean Theorem.

    Pythagorean Theorem [d]

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 5
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right
triangle.

    triangle [d]
    hypotenuse
    altitude [d]

Use special right triangles (30° - 60° and 45° - 45°) to solve problems.

    triangle [d]
    30-60-90 special right triangle
    45-45-90 special right triangle

Define and use the trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) in terms
of angles of right triangles.

    trigonometric function [d]
    sine function [d]
    cosine function [d]
    tangent function [d]
    cosecant function [d]
    secant function [d]
    cotangent function [d]
    right triangle [d]
    angle [d]

Know and use the relationship sin2 x + cos2 x = 1.

    sin2 x + cos2 x = 1 [d]

Solve word problems involving right triangles.

    right triangle

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 6
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Standard 6
Circles

Students define ideas related to circles: e.g., radius, tangent. They find measures of angles, lengths,
and areas. They prove theorems about circles. They find equations of circles.

Find the center of a given circle. Construct the circle that passes through three given points (not in a
straight line).

    circle [d]
    center of a circle [d]
    line

Define and identify relationships among: radius, diameter, arc, measure of an arc, chord, secant, and
tangent.

    diameter
    chord [d]
    arc [d]
    measure of an arc [d]
    secant
    tangent [d]

Prove theorems related to circles.

    circle [d]
    theorem [d]

Construct tangents to circles, and circumscribe and inscribe circles.

    circle [d]
    tangent [d]
    circumscribed circle
    inscribed circle

Define, find, and use measures of arcs and related angles (central, inscribed, and intersections of secants
and tangents).

    angle [d]
    arc [d]
    measure of arc [d]
    central angle
    inscribed angle
    intersection of secants
    intersection of tangents

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 7
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Define and identify congruent, similar, and concentric circles.

    congruent circles
    similar circles
    concentric circles

Define, find, and use measures of circumference, arc length, and areas of circles and sectors. Use these
measures to solve problems.

    circle [d]
    circumference
    arc length
    sector of a circle
    area of a sector of a circle

Find the equation of a circle in the coordinate plane in terms of its center and radius.

    center of a circle [d]
    equation of a circle [d]

Standard 7
Polyhedra and Other Solids

Students describe and make polyhedra and other solids. They describe relationships and symmetries,
and use congruence and similarity.

Describe and make regular and non-regular polyhedra.

    polyhedron [d]
    regular polyhedron
    non-regular polyhedron

Describe the polyhedron that can be made from a given net (or pattern). Describe the net for a given
polyhedron.

    net
    polyhedron [d]

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 8
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Describe relationships between the faces, edges, and vertices of polyhedra.

    face [shape]
    edge [shape]
    vertex
    polyhedron [d]

Describe symmetries of geometric solids.

    symmetries of solids

Describe sets of points on spheres: chords, tangents, and great circles.

    sphere
    chord [d]
    great circle
    tangent [d]

Identify and know properties of congruent and similar solids.

    congruent solids
    similar solids

Standard 8
Mathematical Reasoning and Problem Solving

Students use a variety of strategies to solve problems.

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.

    problem-solving strategy [d]
    guess-and-check [d]
    working backwards [d]

Decide whether a solution is reasonable in the context of the original situation.

    reasonable solution [d]

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 9
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Students develop and evaluate mathematical arguments and proofs.

Make conjectures about geometric ideas. Distinguish between information that supports a conjecture and
proof of a conjecture.

    conjecture
    proof of a conjecture

Write and interpret statements of the form “if – then” and “if and only if.”

    if-then (conditional) statement
    if and only if statement

State, use, and examine the validity of the converse, inverse, and contrapositive of “if – then”
statements.

    converse of “if-then” statement
    inverse of “if-then” statement
    contrapositive of “if-then” statement

Identify and give examples of undefined terms, axioms, and theorems, and inductive and deductive
proof.

    axiom
    theorem [d]
    inductive proof
    deductive proof [d]

Construct logical arguments, judge their validity, and give counterexamples to disprove statements.

    logical argument
    valid argument
    counterexample

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 chart, paragraph, two-
column, and indirect.

    geometric proof
    coordinate proof
    deductive proof [d]
    flow chart proof
    paragraph proof
    two-column proof
    indirect proof

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 10
DRAFT – MATHEMATICS: TERMS AND CONCEPTS FOR GEOMETRY – DRAFT

Perform basic constructions, describing and justifying the procedures used. Distinguish between
constructing and drawing geometric figures.

    construction of geometric figures

[d] = Term or concept is duplicated within or among high school course(s)

PAGE 11

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