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Gateway Regional School District

SCOPE & SEQUENCE

Geometry - Advanced

Based on November 2000 Massachusetts Mathematics Framework – Standards for Geometry

January 2009

of 23

MA

Standards Priority

Curriculum

Benchmarks

Possible Instructional

Strategies

Evidence of Student

Learning (Assessment) Month

TEXTBOOK - Geometry for Enjoyment and Challenge published by McDougal Littell in 1991

UNIT I - Introduction to Geometry G.G.1 Recognize special

types of polygons (e.g.,

isosceles triangles,

parallelograms, and

rhombuses). Apply

properties of sides,

diagonals, and angles in

special polygons; identify

their parts and special

segments (e.g., altitudes,

mid-segments);

determine interior angles

for regular polygons.

Draw and label sets of

points such as line

segments, rays, and

circles. Detect

symmetries of geometric

figures.

Students will KNOW: �How to recognize points, segments, lines,

rays, angles, and triangles

�How to measure segments and angles (in

degrees, minutes and seconds)

�How to classify angles by size or measure

�How to identify midpoints

�How to identify segment and angles bisectors

and trisectors

Students will be able to DO: �Identify and differentiate between segments,

lines and rays

�Recognize and name angles in three different

ways

�Classify angles according to their

size/measure

�Add and subtract angle measurements in

degrees, minutes and seconds

�Understand the definition of midpoints and

its implication

�Understand the definitions of angle and

segment bisectors and trisectors

�Apply midpoints, bisector and trisector

definitions to two column proofs

• Define the terms point, segment, line, ray,

angle, midpoint, angle bisector, trisector, and

triangle

• Demonstrate how to measure segments and

angles

• Work through many examples of measuring

segments and angles

• Work through many examples involving

midpoint and angle bisectors and trisectors

• Student can clearly

define what a segment,

line, ray, angle,

midpoint, angle

bisector, trisector, and

triangle is

• Student can accurately

measure segments and

angles

• Student can correctly

apply the definitions in

this unit to a variety of

examples Sept

G.G.2 Write simple

proofs of theorems in

geometric situations,

such as theorems about

congruent and similar

figures, parallel or

perpendicular lines.

Distinguish between

postulates and theorems.

Students will KNOW: �How to recognize congruent segments and

angles

�How to recognize collinear and non-collinear

points and the concept of betweenness

�How to interpret a diagram

�How to write a simple two-column proof

�How to write a paragraph proofs

�That geometry is based on a deductive

• Define meaning of congruent segments,

collinear points, non-collinear points, inverse,

converse, contrapositive, deductive structure,

and angles

• Explain the proper terminology to use when

constructing a mathematical proof

• Walk through several examples of writing

mathematical proofs

• Explain to the students how one should

• Student can correctly

define and apply the

following terms:

congruent segments,

collinear points, non-

collinear points,

inverse, converse,

contrapositive,

deductive structure, and


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

Use inductive and

deductive reasoning, as

well as proof by

contradiction. Given a

conditional statement,

write its inverse,

converse, and

contrapositive.

structure

�How to identify undefined terms, postulates

and definitions

�How to write an inverse, converse, and

contrapositive when given a conditional

statement

Students will be able to DO: �Understand the use of tic marks in diagrams

�Understand the concept of congruency

through angles and segments with the same

measure

�Able to recognize collinear and non-collinear

points in a diagram

�Understand the concept of betweenness

�Be able to interpret a diagram by assuming:

straight lines and angles, collinearity of points,

betweenness of points, and relative position of

points

� understand that right angles, congruent

segments & angles, and relative size of angles

and segments cannot be assumed from a

diagram

� begin to write a two-column proof matching

a reason for every statement made

�Practice how to reach a conclusion and

convince others of its validity through

paragraph form

�Understand the structure of deductive

reasoning based on definitions, postulates and

theorems

�Understand conditional statements: if a then

b

�Write converses of a � p then p�a

�Write the inverses of "a � p which is not a

� not p"

�Write the contrapositive of “a � p which is

not p � not a”

interpret a geometric diagram through

example

• Walk through several examples of

conditional statements

• Walk through several examples of calculating

angles and segments with algebraic

relationships

• Describe the concept of collinear and non-

collinear points

• Walk through examples of collinear and non-

collinear points

angle

• Students can accurately

calculate angle sums

and conversions

• Students can correctly

use algebraic ratios to

calculate segment ratios


apply the terms of

collinear and non-

collinear points to

mathematical proofs

Oct


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

G.G.15 Draw the results,

and interpret

transformations on

figures in the coordinate

plane, e.g., translations,

reflections, rotations,

scale factors, and the

results of successive

transformations. Apply

transformations to the

solution of problems.

(10.G.9)

Students will KNOW: �How to draw a reflection over a given line

�How to draw a rotation around a given point

Students will be able to DO: �Recognize that a reflection is a mirror image

�Know the difference between clockwise and

counter-clockwise

�Know how to move an object 900, 1800 or

270 o

• Define the terms of reflection and rotation

• Walk through several examples of drawing

reflections over a given line and rotations

centered on a given point

• Students understand

reflection and rotation


draw reflections around

a given line


draw a rotation around a

given point

UNIT II - Basic Concepts and Proofs G.G.2 Write simple







Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and

contrapositive.

Students will KNOW:

�The concept of perpendicularity

�How to write a simple proof

Students will be able to DO:

�Recognize perpendicular lines/segments and

the use of the symbol, ⊥

�Understand that perpendicularity implies

right angles (900)

�Write proofs using the procedures for

drawing conclusions

�Use definitions, theorems or postulates to

justify each statement

• Define the concept of perpendicularity


perpendicular segments

• Look at the implications of perpendicularity

• Show examples of simple formal proofs


mathematical proofs

• Students can define the

concept of

perpendicularity


apply the concept of

perpendicularity to a

mathematical proof


construct simple

mathematical proofs

October

G.G.6 Apply properties

of angles, parallel lines,

arcs, radii, chords,

tangents, and secants to

solve problems.

Students will KNOW:

�Recognize complementary and

supplementary angles and the implication of

their relationships

�How to apply the addition, subtraction,

multiplication and division properties of

angles and segments

• Define complementary and

supplementary angles

• Calculate these angle relationships

• Walk through examples of the addition

and multiplication properties for angles

and segments

• Walk through examples of the

• Students will be able to

differentiate between an

angle’s complement and

its supplement

• Students will calculate

the complements and

supplements of a given


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

�How to apply the transitive and substitution

properties of angles and segments

�Recognize vertical angles

Students will be able to DO: �Understand the definitions of complementary

and supplementary

�Apply the complementary and

supplementary relationships in problem

solving

�Write proofs where students justify

statements by the addition, subtraction,

multiplication and division properties

�Differentiate between the transitive and the

substitution properties

�Use the transitive and substitution properties

as part of their proofs.

�Identify vertical angels in their diagrams

�Use the vertical angles congruency theorem

in their proofs

subtraction and division properties for

angles and segments

• Differentiate between the transitive and

the substitution properties of angles and

segments

• Have students practice applying each of

the properties in their proofs as they

pertain to angles and segments

• Investigate vertical angles and prove their

congruency as based on each being a

supplement to a given angle

• Walk through examples of using that

vertical angles are congruent in a formal

proof

angle

• Students will find the

size of an angle

algebraically based on

angle relationships


apply the addition and

multiplication

properties to their

proofs


apply the subtraction

and division properties

to their proofs

• Students can

appropriately apply

both the transitive and

substitution properties

• Students will recognize

vertical angles and the

fact that they are

congruent in their

proofs

G.G.15 Draw the results,

and interpret

transformations on

figures in the coordinate

plane, e.g., translations,

reflections, rotations,

scale factors, and the

results of successive

transformations. Apply

transformations to the


(10.G.9)

Students will KNOW: �How to draw a reflection over a given line

�How to draw a rotation around a given point

Students will be able to DO: �Recognize that a reflection is a mirror image

�Know the difference between clockwise and

counter-clockwise

�Know how to move an object 900, 1800 or

270 o

• Use a visual model to demonstrate a

reflection over a given line

• Use a visual model to demonstrate a

rotation around a given point

• Students will draw examples of both

reflections and rotations on a coordinate

plane.


draw both reflections

(over a given line) and

rotations (around a

given point)


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

UNIT III - Congruent Triangles G.G.1 Recognize special



parallelograms, and

rhombuses). Apply






mid-segments);

determine interior angles

for regular polygons.

Draw and label sets of

points such as line

segments, rays, and

circles. Detect


figures.

Students will KNOW:

�How to identify medians and altitudes

Students will be able to DO: �Apply the property of medians(that a median

divides the segment into two congruent

segments)

�Apply the property of altitudes (forms right

angles at the points of intersection)

• Define median and altitudes

• Have students investigate the relevance

of these lines to proving triangles congruent


define median and

altitude


use definitions of

medians and altitudes

within a geometric

proof

G.G.2 Write simple







Distinguish between


Use inductive and


well as proof by

contradiction.

Students will KNOW: �How to prove triangles congruent by

applying the SSS, SAS, and ASA postulates

�Apply the principle of CPCTC

�Recognize some basic properties of circles

�Understand why auxiliary lines are used in

proofs

Students will be able to DO: �Analyze the diagrams to determine and use

the appropriate triangle congruency postulates

�Will be able to prove segments and angles

are congruent by recognizing corresponding

parts of congruent triangles

�Use the congruency of radii of a circle in

proofs

• Demonstrate the SAS postulate using a ruler

and protractor

• Have students discover the ASA and the SSS

postulates using a ruler and a protractor

• Define the CPCTC principle

• Have students analyze and write proofs

• Have student draw auxiliary lines as

necessary for their proofs

Students will accurately analyze

and write proofs using SAS, ASA

and the SSS postulates and the

CPCTC principle

November


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

G.G.4 Draw congruent

and similar figures using

compass, straightedge,

protractor, or computer

software. Make

conjectures about

methods of construction.

Justify the conjectures by

logical arguments.

(10.G.2)

Students will KNOW: �How to use a ruler and protractor to draw

congruent triangles

Students will be able to DO: �Draw an angle of a specific measurement

using a protractor

�Draw segments of a specific measurement

using a ruler

�Make conjectures about congruency of

triangles from drawings

• Demonstrate the SAS postulate using a

ruler and protractor

• Have students discover the ASA and the

SSS postulates using a ruler and a

protractor

• Define the CPCTC principle

In addition to standard course, advanced will:

• Have students analyze and write proofs

• Students will accurately

analyze and write

proofs using SAS, ASA

and the SSS postulates

and the CPCTC

principal

• Students will show an

understanding of

applying the concept of

CPCTC by finding

corresponding

angles/sides congruent

after first proving that

the triangles that

contain those

angles/sides are

congruent

• Student will go beyond

CPCTC (i.e. proving an

angle is bisected etc.)

November

G.G.5 Apply congruence

and similarity

correspondences (e.g.,

∆ABC ≅ ∆XYZ) and

properties of the figures

to find missing parts of

geometric figures, and

provide logical

justification. (10.G.4)

Students will KNOW: �Understand what congruent figures are

�How to identify the corresponding parts

�Understand the reflexive property

Students will be able to DO: �dentify congruent geometric figures

�Identify the corresponding parts

�Use the reflexive property in proofs.

• Define congruency and corresponding parts

• Define reflexive property and give examples

of their use

Have students identify congruent figures and

congruent corresponding parts

Students can accurately work with

congruent figures and their

corresponding parts

November





solve problems.

Students will KNOW:

�Recognize some basic properties of circles

Students will be able to DO: �Use the congruency of radii of a circle in

proofs

• Have students recognize that the radii of a

circle are congruent

• Student properly use

congruent radii in their

proofs

November


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

G.G.8 Use the properties

of special triangles (e.g.,

isosceles, equilateral,

30º–60º–90º, 45º–45º–

90º) to solve problems.

(10.G.6)

Students will KNOW: �The various types of triangles and their parts


�Identify triangles according to congruent

sides

�Identify triangles according to angles

measures

�Apply the definitions of the types f triangles

and angles to solve numerical problems and in

proofs

• Define scalene, isosceles, and equilateral

triangles

• Define acute, right, obtuse and equiangular

triangles

• Have students solve problems using different

types of triangles

• Student properly use

congruent radii in their

proofs

November

UNIT IV - Lines in the Plane G.G.2 Write simple







Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and

contrapositive.

Students will KNOW: �How to incorporate the concepts of:

equidistance and perpendicular bisection in

their proofs.


�use definitions, postulates and theorems

related to equidistance and perpendicularity

and perpendicular bisection in a proof.

• Review the definitions of equidistance,

perpendicularity and perpendicular bisection.

• Review the postulates and theorems related to

equidistance, perpendicularity and

perpendicular bisection.

• Apply these definitions, postulates and

theorems to their two-column proofs.

• Students will

demonstrate an

understanding of the

definitions, postulates

and theorems related to

equidistance and

perpendicularity by

using perpendicular

bisectors in their proofs. December


and similarity






provide logical


Students will KNOW: �How to use a detour proof to show

congruencies of triangles or missing

angles/sides.

�How to organize information and draw

diagrams for problems presented in words.

• Demonstrate how to prove a pair of triangles

congruent in order to use a pair of

corresponding parts from those triangles to

prove a different pair of triangles congruent

• Have students look at several examples and

decide which of them require a detour proof

• Have students look for the triangles that

would need to be proven first

• Have students work in pairs to complete a

• Students will show an

understanding of detour

proofs by recognizing

when they do not have

enough information to

prove given figures

congruent. They will

have to prove another

pair of triangles

December


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

Students will be able to DO: �To prove more than one pair of triangles

congruent in order to solve the problem.

�Read the problem, identify the givens,

identify what needs to be proven, draw and

label an appropriate diagram and write a

formal or paragraph proof.

“detour” proof

• Students will practice reading a problem with

no diagram; they will then draw and label a

diagram that depicts the written description

• Once a labeled diagram is drawn, students

will determine a list of the givens and decide

what needs to be proven

• Students will use their diagrams and the

givens to complete their proof

congruent first in order

to use their congruent

corresponding parts for

the given proof.


organize the

information in and draw

appropriate diagrams

for problems presented

in words.





solve problems.

Students will KNOW: �How to show that two angles are right angles

if they are both supplementary and congruent

Students will be able to DO: �Prove perpendicularity by showing angles

are right angles

• Using an algebraic proof, students will

understand that two angles that are both

supplementary and congruent add up to

ninety degrees and therefore are right

angles.

• Students will then apply this relationship

to prove that two lines are perpendicular.

• Students will

demonstrate their ability

to prove that lines are

perpendicular by

finding right angles or

congruent adjacent

angles

December




30º–60º–90º, 45º–45º–


(10.G.6)

Students will KNOW:

�How to apply the equidistance theorems to

prove that triangles are isosceles


�Prove that a triangle is isosceles by using an

equidistance theorem

• Define distance: the distance between two

points is the length of the segment that joins

them.

• Define equidistant: if two points are the same

distance from a third point, they are said to be

equidistant.

• Define perpendicular bisector: a line that both

bisects and is perpendicular to a segment.

• Students will be

proving that a given

triangle is isosceles by

showing that points are

equidistant from the

endpoints of the base.

December

G.G.12 Using rectangular

coordinates, calculate

midpoints of segments,

slopes of lines and

segments, and distances

between two points, and

apply the results to the

solutions of problems.

Students will KNOW: �The midpoint formula

�Understand the concept of slope

�Determine the slope of a line using a formula

�The relationship between the slopes of

parallel and perpendicular lines

Students will be able to DO: �Calculate the midpoint of segments using the

midpoint formula

�Apply the definition of equidistance to solve

problems

�Recognize that a rising line has a positive

• Define midpoint: a point or segment that

divides a segment into two congruent parts.

• Have the students find the midpoint of a line

segment by measuring.

• Have the students use the formula to find the

midpoint of a diagonal line.

x1 + x2

2,y1 + y2

2

• Review the interpretation of a graph having a

slope that is positive, negative, zero and no

slope.

• Have students look at lines drawn on a


tell if a line is

horizontal, vertical, or

diagonal by its slope.

• Students will correctly

determine the slope of a

line by applying the

slope formula.

• Students will

demonstrate an


relationship between

slope and lines being

December


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

slope, a falling line has a negative slope, a

horizontal line has a zero slope and a vertical

line has an undefined slope

�Use the slope formula to determine the slope

of a line.

� Use the slope formula to determine the slope

of a line.

�Calculate slope and then determine if the

lines are parallel, perpendicular or neither

coordinate plane. They will find the slope by

counting how far up/down and how far to the

right the line travels

• Define the slope formula as ∆y

∆x

• Students will use the formula:y2 − y1

x2 − x1

= m

• Students will find the slopes of two or more

lines; they will then determine if the lines are

parallel (same slopes) or perpendicular

(slopes are opposite reciprocals)

parallel or

perpendicular.

UNIT V - Parallel Lines and Related Figures G.G.1 Recognize special



parallelograms, and

rhombuses). Apply






midsegments); determine

interior angles for regular

polygons. Draw and label

sets of points such as line

segments, rays, and

circles. Detect


figures.

Students will KNOW: �How to recognize and name four sided

polygons

�Recognize diagonals

�The names and definitions of special

quadrilaterals

Students will be able to DO: �Differentiate between figures that are

polygons and those that are not polygons

�Name polygons according to the number of

sides

�Name diagonals in polygons

�Name and identify parallelograms,

rectangles, rhombuses, kites, squares,

trapezoids and isosceles trapezoids

• Teacher will show a number of pictures of

figures for students to differentiate a

polygons or not

• Teacher will define the special quadrilaterals

• Students will use prior knowledge and

definitions to identify special quadrilateral

• Students will draw different quadrilaterals

according to definition and use measurement

of segments and angles to identify properties

• Students will compile a list of the properties

of special quadrilaterals


recognize figures as

being a polygon


accurately name each of

the special

quadrilaterals

• Students’ abilities to

discover the unique

properties of specific

quadrilaterals

January

G.G.2 Write simple





Students will KNOW: �The indirect proof procedure

�The various methods to prove that lines are

parallel

�The parallel line postulate

• The teacher will demonstrate an indirect

proof.

• The students will practice an indirect proof

by assuming the opposite of the “to prove” is

true and then find a contradiction to one of

• Students will

demonstrate an

understanding of

indirect proofs by

applying the idea of


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23



Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and

contrapositive.

�The methods to prove that a quadrilateral is a

parallelogram or a rectangle, rhombus, kite,

square, trapezoid or an isosceles trapezoid

Students will be able to DO: �Use contradiction to solve indirect proofs

�Prove that lines are parallel by showing

alternate interior, corresponding, alternate

exterior angles are congruent or that same side

interior, same side exterior angles are

supplementary or that lines are perpendicular

to the same line

�To use the properties of the special

quadrilaterals in their proofs

the givens.

contradiction


identify all angles

associated with parallel

lines to accurately use a

variety of methods to

prove that lines are

parallel.

January


and similarity






provide logical


Students will KNOW: �The properties of each of the special

quadrilaterals

�The methods to prove that a quadrilateral is a

parallelogram or a rectangle, rhombus, kite,

square, trapezoid or an isosceles trapezoid

Students will be able to DO: �Identify the properties and use these

properties to solve problems

�To use the properties of the special

quadrilaterals in their proofs

• Teacher will demonstrate the use of

numerical and algebraic calculations to find

missing parts of figures

• Students will independently practice finding

missing parts of figures

• Students will work in groups to apply the

properties of special quadrilaterals to their

proofs

• Students will independently prove that

figures are special quadrilaterals

• Students will show

mastery of using

numerical and algebraic

calculations to find

missing parts of a figure


apply the properties of

the special

quadrilaterals to their

proofs


prove that figures are

special quadrilaterals

January





solve problems.

Students will KNOW:

�How to identify pairs of angles formed by a

transversal cutting two parallel lines

�How to prove that angles are congruent

associated with parallel lines.

Students will be able to DO: �Be able to identify alternate interior angles,

corresponding angles, alternate exterior

angles, same side interior and same side

exterior angles

• Define alternate interior angles,

corresponding angles, vertical angles,

alternate exterior angles and same side

interior and same side exterior angles.

• Students will explore the ways in which to

prove lines parallel by applying the above

definitions.

• Students will apply theorems relating interior

and exterior angles on the same side of the

transversal as being supplementary.

• Students will apply the theorem that states if


identify specific pairs of

angles associated with

parallel lines

January


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

�Prove that corresponding, alternate exterior

angles, and alternate interior angles are

congruent or that same side interior, same side

exterior angles are supplementary or that lines

are perpendicular to the same line

two coplanar lines are perpendicular to a

third line they are parallel.

• Students will draw a pair of parallel lines that

are cut by a transversal and they will identify

each of the specific pairs of resulting angles.

UNIT VI - Polygons G.G.1 Recognize special



parallelograms, and

rhombuses). Apply






midsegments); determine

interior angles for regular

polygons. Draw and label

sets of points such as line

segments, rays, and

circles. Detect


figures.

Students will KNOW: �That the sum of the three angles of a triangle

equals 180°

�How to find the number of diagonals for a

given polygon

�The definition of a regular polygon

�How to find the measure of each exterior

angle of a regular polygon

Students will be able to DO: �Use the formulas to calculate the sum of the

interior and exterior angles of a given polygon

�Use the formula to calculate the number of

diagonals for a given polygon

�Apply the definition of regular polygons

�Find the measure of each exterior angle of a

regular polygon knowing the number of sides

�Find the number of sides of a regular

polygon knowing the measure of the exterior

angle

• Teacher will use the parallel postulate to

demonstrate the sum of the angles of a

triangle are 180 degrees

• Students will work in small groups drawing

polygons and counting the number of

diagonals form one vertex and then total

number of diagonals. After doing this for

several polygons they should be able to write

an equation for the total number of diagonals

for any polygon.

• Given the definition of regular polygon, the

students will to calculate the number of

degrees in each interior angle and then use

supplementary angles to find the number of

degrees in an exterior angle.

• Students will be given a formula to find the

number of degrees in each exterior angle.

• The students will apply the same formula to

find the number of sides of a regular given

the degrees in one of its exterior angles

• Teacher will lead the students to prove the

no-choice theorem and the angle, angle, side

theorem

• Teacher will lead the students to prove

congruency between triangles and to find the

missing angle by applying the no-choice

theorem and the angle, angle, side theorem.


calculate a missing

angle of a triangle based

on the total number of

degrees being 180°


find the number of

diagonals for any given

polygon by using the

correct formula.


calculate the size of an

exterior angle given the

number of sides or the

number of sides when

given the size of the

exterior angle


accurately apply the no-

choice theorem and

angle, angle, side

theorem to their proofs.

February

G.G.2 Write simple






Students will KNOW: �The no-choice theorem

Angle, Angle, Side theorem

Students will be able to DO: �To prove that if two angles of one triangle

• Teacher will use the parallel postulate to

demonstrate the sum of the angles of a

triangle are 180 degrees

• Given the definition of regular polygon, the

students will to calculate the number of

degrees in each interior angle and then use


calculate a missing

angle of a triangle based

on the total number of

degrees being 180°


February


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23


Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and

contrapositive.

are congruent to two angles of another

triangle, the third angles of each are congruent

to each other

�Prove that two triangles are congruent by

showing that two angles and the non-included

side of one triangle are congruent to the

corresponding sides of the other

supplementary angles to find the number of

degrees in an exterior angle.

• Students will be given a formula to find the

number of degrees in each exterior angle.

• The students will apply the same formula to

find the number of sides of a regular given

the degrees in one of its exterior angles

find the number of

diagonals for any given

polygon by using the

correct formula.


calculate the size of an

exterior angle given the

number of sides or the

number of sides when

given the size of the

exterior angle.

G.G.7 Solve simple

triangle problems using

the triangle angle sum

property, and/or the

Pythagorean theorem.

(10.G.5)

Students will KNOW:

�The triangle angle sum property

�The exterior angle of a triangle property

�The midline theorem

Students will be able to DO: �Use the sum of the measures of the interior

angles of a triangle is 1800 to find the measure

of a missing angle

�Use the measure of the exterior angle of a

triangle is equal to the sum of the two remote

interior angles to solve problems

�Find missing angles and the length of the

third side of a triangle by applying the midline

theorem

• The teacher will lead the students to prove

the midline theorem

• Students will practice applying the midline

theorem to their proofs

• Students will know

when it is appropriate to

apply the midline

theorem to their proofs.

February

UNIT VII - Similar Polygons G.G.2 Write simple







Distinguish between


Use inductive and


well as proof by

Students will KNOW:

�How to prove triangle similar

�The concept of similarity to establish the

congruence of angles and proportionality of

segments

Students will be able to DO: �Prove that triangles are similar by AA, SAS

similarity, SSS similarity theorems

�Prove that angles are congruent and

segments are proportional by establishing the

similarity of two triangles

• Define similar: figures with the same shape

but not necessarily the same size.

• Have students look around the room to

discover shapes that appear to be similar

• Similar polygons can either be dilations

(enlargements) or reductions.

• Have students look at examples of figures

that are either a dilation or a reduction.

• Define similar polygon: the corresponding

angles are congruent and the corresponding

sides are proportional


recognize similar

polygons


correctly find the

missing sides of similar

polygons


correctly identify figure

that have been dilated

or reduced

March


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23



write its inverse,

converse, and

contrapositive.


and similarity






provide logical


Students will KNOW: �How to work with ratios and with

proportions

�The definition and characteristics of similar

polygons

�The sum of the measures of the interior

angles of a triangle is 1800

�The side splitter theorem

�The parallel line theorem

�The angle bisector theorem


�Solve proportion examples

�Find the geometric mean/mean proportional

�Find missing terms of proportions

�Apply the definition of similar polygons to

find a missing length of a side and the

measure of an angle

�Set up a proportions to find the lengths of

segments

• Have students practice finding a missing side

of similar figures by using proportions

• Teacher explanation of the theorem that

states: the ratio of the perimeters of two

similar polygons equals the ratio of any pair

of corresponding sides

• The teacher will define ratio as the quotient

of two numbers

• Show that slope can be shown as a ratio:

rise/run

• Define proportion: an equation between two

ratios, using a colon, or as a product.

• Student will practice solving proportions by

multiplying opposite corners and dividing by

the coefficient of the resulting variable term.

• The teacher will show that “the product of the

means is equal to the product of the

extremes” is another way of saying how the

class has been solving proportions.

• Use the four terms:

1st proportional 3rd proportional

2nd proportional 4th proportional

• Define arithmetic mean as an average

• Define geometric mean/mean proportional:

where the means are the same number

• Together the class will prove that if a line is

drawn parallel to one side of the triangle; it

divides the other two sides proportionally

• Students will be given a sheet with three

parallel lines drawn that have been cut by a

transversal; they will measure the resulting

segments.

• They will draw a second transversal and

again measure the resulting segments. They


find either a missing

side or perimeter given

the other information


calculate arithmetic

ratios and solve

proportions.


identify and solve for

the first, second, third

and fourth proportional


apply the concept of the

geometric mean/mean

proportional to solve

examples.


apply the concept of

similarity to polygons

and then find a missing

side by using

proportions.


apply the side-splitter

theorem, the angle

bisector theorem and

the theorem that states

that transversals are

divided proportionally.

• Students will prove

similarity between two

triangles by applying

the methods of AAA,

March


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

will form a generalization: the parallel lines

divide the transversals proportionally

• The student will work as a group to prove

that if a ray bisects an angle of a triangle, it

divides the opposite sides into segments that

are proportional.

• Students will review the definition of similar

and apply the AAA postulate to prove that

triangles are similar.

• Students will apply the no choice theorem to

understand that only two angles must be

congruent to have similar triangles.

• Students will read examples that show how to

prove that triangles are similar by using the

SSS similarity and SAS similarity.

• Students will practice proving triangles

similar using a variety of methods.

AA, SSS ~ , and SAS ~

UNIT VIII - Right Triangle Properties G.G.5 Apply congruence

and similarity






provide logical


Students will KNOW: �how to simplify radicals

�how to solve quadratic equations

�Relationships between the parts of a triangle

when the altitude is drawn to the hypotenuse

Students will be able to DO: �simplify radical expressions

�Solve quadratic equations either by using the

quadratic formula or by factoring

�Determine similarity of triangles when the

altitude is drawn to the hypotenuse

�Find the measures of segments using a

proportion

• Students will review how to simplify radical

expressions

• Students will review factoring quadratics and

the will solve quadratic equations by

factoring

• Students will draw a right triangle with an

altitude to the hypotenuse. They will make

conjectures based on this drawing.

• The group will prove that the triangles

formed by an altitude to the hypotenuse are

similar.

• The class will prove that one leg is the mean

proportional to the hypotenuse and segment

of the hypotenuse adjacent to it.

• The class will prove that the altitude is the

mean proportional to the hypotenuse and

segment of the hypotenuse adjacent to that

leg.

• Students will practice finding the lengths of

the missing segments by applying these

theorems.


accurately simplify

radicals


solve quadratic

equations both by

factoring and by using

the quadratic formula

• Students prove triangles

formed by an altitude to

a hypotenuse are similar

• Students will use the

mean proportional

theorems to find the

length of either a leg or

a segment of the

hypotenuse

April


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

G.G.7 Solve simple

triangle problems using

the triangle angle sum

property, and/or the

Pythagorean theorem.

(10.G.5)

Students will KNOW: �Pythagorean theorem and its converse

�Recognize groups of whole numbers known

as Pythagorean triples

Students will be able to DO: �Apply the Pythagorean theorem to find the

length of a missing side

�Determine if a triangle is a right triangle by

squaring the measures of the three sides

�Determine the missing side of a right triangle

using a Pythagorean triple

• Students will investigate to find the

relationship between the sum of the square of

the legs of a right triangle and the square of

the hypotenuse.

• Students will look at the physical

demonstration of the Pythagorean

relationship (on-line)

• Students will practice using the Pythagorean

relationship to find the lengths of either leg or

the hypotenuse.

• Students will the converse of this theorem to

prove tat the triangles given are indeed right

triangles.

• Students will recognize the families of whole

numbers known as the Pythagorean triples

• Students will practice solving for the missing

side of a right triangle by using the

Pythagorean triples or their multiples

• Students will use

radicals and factoring as

hey apply to the

Pythagorean theorem

and to the altitude on

the hypotenuse theorem

• Students will use the

Pythagorean triples as a

shortcut to solve for the

missing sides April




30º–60º–90º, 45º–45º–


(10.G.6)

Students will KNOW: �The ratio of the side lengths of a 30-60-90

triangle

�The ratio of the side lengths of a 45-45-90

triangle

Students will be able to DO: �Use the ratios of a 30-60-90 and a 45-45-90

triangle to calculate the missing sides

• Students will draw a 30°, 60°, 90° triangle;

they will measure the shortest side and the

hypotenuse. They should notice that the

hypotenuse is always twice as big as the

shortest side.

• They should apply the Pythagorean theorem

to find the length of the third side (in

simplified form); it will always be x√3.

• Following a similar procedure the students

will investigate the lengths of the sides of a

45°, 45°, 90° triangle; they should discover

that the legs are the same and that the

hypotenuse is always the length of the leg

times √2.


find the lengths of

either a leg or a

hypotenuse of a

“special” right triangle

by using the discovered

relationships rather than

having to go through

using the Pythagorean

theorem.

April

G.G.9 Define the sine,

cosine, and tangent of an

acute angle. Apply to the


Students will KNOW: �The three basic trigonometric relationships

�How to use trigonometric ratios to solve right

triangle problems

Students will be able to DO: �Determine the sine, cosine and tangent of a

• Students will do an exploratory activity to

discover the three basic trigonometric

relationships,

• Students will practice writing the sine, cosine

and tangent for many given triangles as a

fraction.

• Students will learn how to use the table of


correctly recognize

which trig function they

need to use for a given

example.


find the missing side or

April


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

given triangle

�Use trigonometric ratios to calculate the

measures of missing sides and angles

trig ratios.

• Students will be given a right triangle with a

side and an angle labeled. They will have to

find the missing part by applying a

trigonometric function.

• Students will expand this to problems where

they must first draw a diagram, label the

given parts and find a missing part by

applying the trigonometric relationships.

angle by using the trig

function.


apply these functions to

answer word problems.

G.G.12 Using rectangular

coordinates, calculate

midpoints of segments,

slopes of lines and

segments, and distances

between two points, and

apply the results to the

solutions of problems.

Students will KNOW:

�The distance formula to compute the lengths

of segments in the coordinate plane


�Calculate the distance between two points

• Students will look at a right triangle drawn on

a coordinate plane. They will label the

points. They will find the vertical and

horizontal lengths by counting spaces and

they will then use the Pythagorean theorem to

find the length of the hypotenuse.

• Based on the above activity students will try

to discover the length of a hypotenuse

without using the Pythagorean theorem.

They will be looking toward the discovery of

the distance formula.

• Once they have found the formula, students

will be practicing finding the lengths of any

side by using the distance formula:

(x2 − x1)2 + (y2 − y1)

2

• Students will apply this formula to a variety

of situations

•


evaluate the distance

between two points and

they will apply this

formula to a variety of

situations

April

G.M.4 Describe the

effects of approximate

error in measurement and

rounding on

measurements and on

computed values from

measurements. (10.M.4)

Students will KNOW: �Rounding before calculations will skew the

results

Students will be able to DO: �Recognize inaccurate results due to rounding

errors or rounding before calculating

• Half the students will be asked to solve an

example without any rounding until they

complete the examples; the other half will

round each time they have decimals within

their work.

• Upon looking at the results of this

investigation, students will have found that if

one were to round their numbers within the

problem rather than waiting to they have an

answer will significantly change the results of

the answers.

• Students will be judged

on their understanding

of the effects of

rounding with the work

they do using the

Pythagorean theorem

and with applying the

distance formula.

April


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

UNIT IX - Circles G.G.2 Write simple







Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and contra

positive.





solve problems.

Students will KNOW:

�That a radius is a perpendicular bisector of a

chord

�If a radius is perpendicular to a chord then it

bisects the chord

�If a radius of a circle bisects a chord the is

not a diameter, then it is perpendicular to that

chord

�The perpendicular bisector of a chord passes

thorough the center of the circle

�The definitions of the parts of a circle

Understand the relationship between diameter,

radii and chords

�The definitions of a major arc, minor arc and

semicircle

�The definition of a central angle

�The measure of a minor arc or a semicircle is

the same as the measure of the central angle

that intercepts the arc

�The relations of congruent arcs, chords, and

central angles

�The definition of secant, tangent, tangent

segment, secant segment, tangent circles,

externally tangent, internally tangent, common

Internal and external tangents

�A tangent line is perpendicular to a chord

�A line perpendicular to a radius at its outer

endpoint is a tangent

�Two tangents drawn from a point to a circle

are congruent

�The common tangent procedure

�The definitions of angles related to a circle

which include central, inscribed and tangent-

chord, chord-chord, secant-secant, secant-

tangent, and tangent-tangent angles

�The theorems establishing the measures of

angles related to a circle

�The definition of inscribed and

• Students will read the radii/chord

theorems and prove the theorems based

on previous knowledge

• Demonstrate the use of the radii/chord

theorems in solving proofs

• Walk through some numerical problems

solved by using the radii/chord theorems

• Define major arc, minor arc, semicircle

and central angle

• Define the measure of a minor arc or

semicircle as the measure of the central

angle that intercepts the arc and a major

arc as being 360° minus the measure of

the minor arc with the same endpoints

• Students will use protractors and

compasses to discover the relationship

between congruent arcs, chords, and

central angles

• Define tangent, secant, tangent segment

and secant segment

• Demonstrate the use of the Two-Tangent

Theorem

• Explain the tangent line postulates

• Students will draw tangent circles in

order to discover how circles can be

tangent (i.e. internally and externally)

• Walk through the common tangent

procedure and let students discover the

use of the Pythagorean theorem in order

to complete the procedure

• Define the angles related to a circle-

inscribed, tangent-chord, chord-chord,

secant-secant, secant-tangent, and

tangent-tangent

• Students will determine an angle- arc

summary according to various theorems

• Introduce inscribed and circumscribed


use the radii/chord

theorems in order to

solve numerical

problems and to

complete proofs

• Students can identify

minor arcs, major arcs,

semi circles, and central

angles


solve numerical

problems and complete

proofs using the

theorems relating arcs,

central angles and

chords


identify tangents,

secants, tangent

segments, secant

segments


circles as internally or

externally tangent

• Students can apply the

two tangent theorem


apply the common

tangent procedure to

solve problems


the angles related to a

circle (i.e. inscribed,

tangent-chord etc.)

• Students can solve

problems and complete

proofs associated with

May


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

circumscribed polygons

�The power theorems

�How to find arc length

Students will be able to DO: �Identify the characteristics of circles, chords,

radii, and diameters

�Recognize the relationship between radii and

chords

�Solve numerical problems using the

relationship of radii, diameters and chords

(G.G.6)

�Write proofs using the relationship between

radii, diameters and chords (G.G.2)

�Apply the relationship between congruent

chords of a circle in doing numerical problems

(G.G.6) and in writing proofs (G.G.2)

�Identify different types of arcs

�Determine the measure of an arc and

recognize congruent arcs (G.G.6)

�Relate congruent arcs, chords and central

angles

�Apply central angle/chord, central angle/arc,

and chord/arc theorems in solving numerical

problems (G.G.6)

�Apply the central angle/chord, central

angle/arc, and chord/arc theorems in writing

proofs (G.G.2)

�Identify secant and tangent lines and

segments

�Distinguish between two types of tangent

circles

�Apply the common tangent procedure

(G.G.6)

�Solve problems using the angle arc theorems

�Solve problems related to inscribed and


�Solve problems using the power theorems

(G.G.6)

�Find arc lengths based on arc measure

polygons and have students draw several

examples of each

• Students will drawn inscribed

quadrilaterals and measure the angles to

discover that the opposite angles are

supplementary

• Students will draw parallelograms in a

circle to discover that it must be a

rectangle

• Students will draw two chords in a circle

and with the help of the teacher will use

measurement and multiplication in order

to discover the Chord-Chord Power

Theorem. The students will then do the

same on their own for the Tangent-Secant

Power Theorem and the Secant-Secant

Power Theorem

• Students will develop the formula for

finding the length of an arc [length of an

arc PQ=(mPQ÷360)πd] by recognizing

the length of an arc being a fractional part

of a circle’s circumference determined by

the arc’s measure

•

angle related to a circle

• Students can recognize

inscribed and


• Students can apply the

relationship between the

opposite angles of an

inscribed quadrilateral


the characteristics of an

inscribed parallelogram


solve problems

concerning inscribed

and circumscribed

polygons


apply the power

theorems


determine the length of

an arc


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

UNIT X - Area G.M.1 Calculate

perimeter, circumference,

and area of common

geometric figures such as

parallelograms,

trapezoids, circles, and

triangles. (10.M.1)

Students will KNOW:

�Understand the concept of area

�Find the area of rectangles, squares and

related irregular figures

�Determine the area of a variety of polygons

�Determine the area of a circle and portions of

a circle

Students will be able to DO: �Define area

�Estimate areas of irregular figures by

determining the approximate number of square

units

�Find the area of polygons by applying the

given formulas.

�Find the area of circles, sectors and segments

of circles by applying formulas

• Define area

• Students should understand that:

• Every closed region has an area

• If two closed figures are congruent, then

their areas are equal

• If two closed regions share a common

boundary, then the area of the entire

figure is the sum of the individual areas.

• Students will practice using the formulas

for finding the area of: rectangles,

squares, parallelograms, and triangles.

• Students will need to correctly identify

the base and the heights of the

parallelograms and triangles.

• Students will be able to use either the

radius or the diameter to find the area of

the circle.

• They will find the area of a sector by

multiplying the area of the circle times

the fractional part of the circle (i.e. if an

arc measures 600, then the sector is 600 ÷

3600 or 1/6 of the circle.

• Students will use three different methods

for finding the area of a trapezoid.

• Breaking the trapezoid into three figures

• Using the formula

• Using the product of the average of the

bases and the height.

• Once they have practiced all three they

may determine the one that works best

for them.

• Students will recognize radii vs.

apothems.

• Students will apply three theorems

related to area:

• If two figures are similar, then the ratio of

their areas equals the square of the ratio

• Students will show that

they understand the

definition of area and its

implications


identify the bases and

the heights in order to

correctly apply the

formulas for areas of

rectangles, squares,

parallelograms, and

triangles.


use either the radius or

diameter to find the area

of circles and they will

be able to find the area

of a sector of a circle.

• Students will

successfully use the

formulas related to

similar figures and to

regular polygons.

• Students will

demonstrate that they

can use a variety of

methods to find the area

of both trapezoids and

kites.

May


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

of corresponding segments.

• A eq∇ = s

2

43

• A reg.poly. = 1

2ap where a is the length of

the apothem and p is the perimeter.

• Students will find the area of a kite by

dividing the kite into isosceles triangles

and using the area of a triangle formula

• Students will use an alternate method for

finding the area of a kite: area equals half

the product of the diagonals.

G.M.3 Relate changes in

the measurement of one

attribute of an object to

changes in other

attributes, e.g., how

changing the radius or

height of a cylinder

affects its surface area or

volume. (10.M.3)

Students will KNOW: �How to relate the change of one attribute to

the overall change in the area of a figure

Students will be able to DO: �Increase or decrease the size of the length,

width or height etc. and calculate the overall

effect this will have on the area of reduction or

the dilation of a figure

• Students will continue to practice finding

areas of all figure and they will calculate

the overall effect changing the size of an

attribute has on the area of the new figure

• Students will show that

they understand the

relationship between

changing the size of an

attribute and the overall

change it will have on

the areas of the given

figures.

May

UNIT XI- Surface Area and Volume G.G.16 Demonstrate the

ability to visualize solid

objects and recognize

their projections and

cross sections. (10.G.10

Students will KNOW: �To recognize and draw various three

dimensional figures

Students will be able to DO: �To name the various solid (three

dimensional) figures

�To draw representations of various solid

figures

�Visualize the projections and cross sections

of various solid figures

• Students will use visual models of the

three dimensional figures to understand

what is meant by the base of the figure,

the lateral edges, lateral faces.

June

G.M.2 Given the

formula, find the lateral

area, surface area, and

volume of prisms,

Students will KNOW:

�How to derive and apply formulas for lateral

area, surface area and volume of various

geometric figures

• Students will find the lateral surface area

of any prism by finding the sum of the

areas of the lateral faces.

• Students will find the total surface area


choose the appropriate

formula and to

accurately use the

June


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

pyramids, spheres,

cylinders, and cones, e.g.,

find the volume of a

sphere with a specified

surface area. (10.M.2)


�Students will derive and apply formulas for

lateral area and surface area of prisms,

pyramids, circular solids

�Students will derive and apply formulas for

volume of prisms, cylinders, pyramids, cones

and spheres

by first finding the lateral surface area

and then adding the sum of each of the

figures bases.

• Students will work with cylinders, cones,

and spheres to determine their respective

formulas for lateral and/or total surface

areas.

• Students will practice applying the

correct formulas to find the lateral and/or

total surface areas of cylinders, cones,

and spheres.

• Volume will be defined as the measure of

the space enclosed by a solid.

• By looking at a physical representation of

a rectangular prism, the students will

come to understand that the volume is

simply the area of the base times the

height of the prism.

• Students will translate this understanding

to other prisms and to a cylinder.


for the volume of prisms and cylinders.

• By looking at physical representations of

a pyramid and a prism with the same base

and height, the students will understand

that the volume of this pyramid is one-

third that of the volume of the prism.


for the volumes of pyramids and cones.

• Students will use the formulas for the

volume of the sphere.

formulas for lateral

surface area and total

surface areas for prisms,

cylinders, cones, and

spheres.

• Students will

demonstrate an


concept of volume.


apply the formulas for

volume of prisms,

cylinders, pyramids and

cones.

G.M.3 Relate changes in

the measurement of one

attribute of an object to

changes in other

attributes, e.g., how

changing the radius or

height of a cylinder

affects its surface area or

Students will KNOW:

�How to relate the change of one attribute to

the overall change in the surface area or

volume of a figure

Students will be able to DO: �Students will increase or decrease the size of

the length, width or height etc. and calculate

• Students will continue to practice finding

volumes of all figures and they will

calculate the overall effect changing the

size of an attribute has on the volume of

the new figure.

Students will show that they

understand the relationship

between changing the size of an

attribute and the overall change it

will have on the areas of the given

figures.

June


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

volume. (10.M.3) the overall effect this will have on the surface

area or volume of the reduction or the dilation

of a figure.

G.M.4 Describe the

effects of approximate

error in measurement and

rounding on

measurements and on

computed values from

measurements. (10.M.4)

Students will KNOW: �Understand the effect of a rounding error on

the overall computed surface areas and

volumes


�To see that rounding a measurement too soon

or by making an error in rounding can greatly

affect the overall calculations when finding

surface areas and volumes

• Different students will use the same

formulas with various decimal values

(tenth, hundredth, thousandth). Upon

investigating, the students should

recognize rounding to different places

would have an effect on the answer to

surface area and volume examples.

Within their work on surface

areas and volumes, the students

will know when to round and to

what decimal value.

June

UNIT XII- Lines and Planes in Space G.G.2 Write simple

proofs of the theorems in






Distinguish between


Use inductive and


well as proof by



write its inverse,

converse, and contra

positive.




tangents and secants to

solve problems

Students will know: �Three non-collinear points determine a plane

(G.G.2)

�A line and a point not on a line determine a

plane (G.G.2)

�Two intersecting lines determine a plane

(G.G.2)

�Two parallel lines determine a plane (G.G.2,

G.G.6)

�If a line intersects a plane not containing it,

then the intersection is exactly one point

(G.G.2, G.G.16)

�If two planes intersect, their intersection is

exactly one line

�If a line is perpendicular to two distinct lines

that lie in a plane and that pass through its

foot, then it is perpendicular to the plane

(G.G.2, G.G.16)

�A line and a plane are parallel if they do not

intersect (G.G.2, G.G.6)

�Two planes are parallel if they do not

intersect (G.G.2, (G.G.6)

�If a plane intersects two parallel planes, the

lines of intersection are parallel (G.G.2,

G.G.6, G.G.16)

• Students will review the definition

of a plane (sec. 4.5) and all

conditions of points and lines in a

plane

• Students will review coplanar and

noncoplanar

• Define the foot of a line with respect

to a plane and have students draw

some examples

• Explain the postulate that three

noncollinear points determine a

plane and have students demonstrate

• Have students investigate to

discover the three other ways to

determine a plane which are three

theorems –a line and a point not on a

line, two intersecting lines, two

parallel lines

• Demonstrate the two postulates

concerning lines and planes

• Define perpendicularity of a line and

a plane

• Explain the theorem that states if a

line is perpendicular to two distinct

lines that lie in a plane and that pass


four ways to determine

a plane


answer questions by

applying postulates and

theorems related to

planes, parallel lines

and perpendicular lines

• Students can complete

proofs by applying

theorems and postulates

related to planes,

parallel lines, and

perpendicular lines

June


SCOPE & SEQUENCE

Geometry - Advanced


January 2009

of 23

G.G.16 Demonstrate the

ability to visualize solid

objects and recognize

their projections and

cross sections

�The properties relating parallel lines and

planes (G.G.2, G.G.6)

Students will do: �Understand basic concepts relating to planes

�Use the four ways to determine a plane in

completing proofs

�Apply the basic theorem concerning the

perpendicularity of a line and a plane

�Complete proofs suggested from three

dimensional diagrams

�Recognize lines parallel to planes, parallel

lines and skew lines

�Apply properties relating parallel lines and

planes

through its foot, then it is

perpendicular to the plane

• Students will determine how a line

and a plane are parallel by trial and

error

• Students will determine how two

planes are parallel by trial and error

• Define two skew lines as

noncoplanar

• Prove the theorem if a plane

intersects two parallel planes, the

lines of intersection are parallel

• Students will discuss the validity of

the following properties of parallel

lines and planes that are presented

without proof:

• If two planes are perpendicular to the

same line, they are parallel 2) If a line is

perpendicular to one of two parallel

planes, it is perpendicular to the other 3)

If two planes are perpendicular to the

same plane, they are parallel to each

other 4) If two planes are parallel to the

same plane, they are parallel to each

other 5) If a plane is perpendicular to one

of two parallel lines, it is perpendicular to

the other as well

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