GRAPHIC ORGANIZER: ANGLES IN A CIRCLEimages.pcmac.org/SiSFiles/Schools/GA/DouglasCounty/ChapelHillHigh/...GRAPHIC ORGANIZER: ANGLES IN A CIRCLE At the CENTER of the circle. ON the

GRAPHIC ORGANIZER:

ANGLES IN A CIRCLE

At the CENTER

of the circle.

ON

the circle.

INSIDE

the circle.

OUTSIDE

the circle.

Two Radii

Two Chords, or

A Chord and a Tangent, or

A Chord and a Secant.

Two chords Two Secants, or

A Secant and a Tangent, or

Two Tangents.

The measure of the

Intercepted Arc

Half the measure of the

Intercepted Arc

Half the SUM of the measures

of the Intercepted Arcs.

Half the DIFFERENCE of the

measures of the intercepted

arcs.

The angle EQUALS

The angle is MADE BY

The VERTEX is located

GRAPHIC ORGANIZER:

SEGMENTS IN A CIRCLE

The VERTEX is located

INSIDE OUTSIDE

the circle the circle

Two chords Two Secant Segments A Tangent and a Secant Segment

GRAPHIC ORGANIZER:

CLASSIFY QUADRILATERALS

NO Pairs of

Opposite Sides

Are Parallel

Exactly ONE Pair of

Opposite Sides

Are Parallel

BOTH Pair of

Opposite Sides

Are Parallel

GRAPHIC ORGANIZER:

POLYGONS – DEFINITION

GRAPHIC ORGANIZER:

SOME PROPERTIES

PROPERTY

ADDITION

MULTIPLICATION

COMMUTATIVE

A + B = B + A

A ! B = B ! A

ASSOCIATIVE

A + ( B + C ) = (A + B ) + C

A ! ( B ! C ) = ( A ! B ) ! C

IDENTITY

A + 0 = A

A ! 1 = A

INVERSE

A + B – B = A

A ! B ! (1/B) = A

PROPERTIES OF EQUALITY

Addition Property of Equality:

If A = B, Then A + C = B + C

Subtraction Property of Equality

If A = B, Then A – C = B – C

Multiplication Property of Equality:

If A = B, Then A ! C = B ! C

Division Property of Equality:

If A = B and C " 0, then (A / C) = (B / C)

REFLEXIVE

A = A

DISTRIBUTIVE

A ! ( B + C ) = A ! B + A ! C

TRANSITIVE

If A = B And If B = C, Then A = C

Classifying Triangles

Find point C that makes !ABC :

a) equilateral

b) isosceles

c) right

d) acute

e) obtuse

f) scalene

Triangle Classifications

Given: AB is a side of !ABC

Draw the loci of all points for C so that

!ABC is:

a) equilateral

b) isosceles

c) right

d) acute

e) obtuse f) scalene

We Classify Triangles

By Angles

Obtuse – one angle is obtuse

Right – one angle is right

Acute – all 3 angles are acute

Equilangular – All 3 angles are 60°

AND

By Sides

Scalene – No sides are congruent

Isosceles – At leat two sides are congruent

Equilateral – All 3 sides are congruent

Problem:

CLASSIFYING TRIANGLES BY ANGLES AND SIDES

ISOSCELES

(at least two congruent sides) SCALENE

(no congruent sides)

RIG

HT

(one

rig

ht

ang

le)

45°45°

OB

TU

SE

(one

ob

tuse

an

gle

)

>90°

>90°

>90°

AC

UT

E

(all

acu

te a

ng

les)

<90°

<90°

EQUILATERAL (all congruent sides)

EQ

UIA

NG

UL

AR

(a

ll 6

0°

ang

les)

60° 60°

60°

Three angle-side combinations make no sense: Equiangular – Scalene Obtuse – Equilateral Right -- Equilateral

GRAPHIC ORGANIZER:

CLASSIFY NUMBERS

REAL NUMBERS

RATIONAL NUMBERS: … -1, … - !, … 0, … !, … 1, … (ratio of integers, no zero in the denominator)

INTEGERS: … -3, -2, -1, 0, 1, 2, 3, …

WHOLE NUMBERS: 0, 1, 2, 3, …

NATURAL (COUNTING) NUMBERS: 1, 2, 3, …

IRRATIONAL NUMBERS: (non-repeating decimals – cannot be written as a ratio of two integers)

Transcendental (can be multiplied by another irrational

number to make it rational): example 2 , 3 etc…

Non-Transcendental

Example: !, e …

IMAGINARY NUMBERS

Example: !1

GRAPHIC ORGANIZER:

ANGLE RELATIONSHIPS

CORRESPONDING ANGLES

ALTERNATE INTERIOR ANGLES

ALTERNATE EXTERIOR ANGLES

VERTICAL PAIR

Adjacent

Non-Adjacent

SAME-SIDE INTERIOR

SAME SIDE EXTERIOR

Adjacent (LINEAR PAIR)

Non-Adjacent

CONGRUENT:

!1" !2

COMPLEMENTARY:

!5 +!6 = 90°

SUPPLEMENTARY:

!3+!4 = 180°

Angle-Angle Similarity Postulate

(AA Sim)

If two angles of one triangle are congruent to two angles of a second triangle, then the two triangles are similar.

A

C

B

D

F

E

!ABC " !DEF

Extension problem: How would you draw two NON-congruent triangles that have 5 congruent parts? 8

PROVING TRIANGLES CONGRUENT

A

C

B

D

F

E

AB ! DE !A " !D

BC ! EF !B " !E

AC ! DF !C " !F

!ABC " !DEF CPCTC: Corresponding Parts of Congruent

Triangles are Congruent. 1

Side-Side-Side Triangle Congruence Postulate

(SSS)

If the sides of one triangle are congruent to the sides of second triangle, then the two triangles are congruent.

A

C

B

D

F

E

This is one of 4 “shortcuts” to prove triangles congruent that are shown in this brochure. 2

Side-Angle-Side Triangle Congruence Postulate

(SAS)

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

A

C

B

D

F

E

Once you prove two triangles are congruent, you can justify that all the corresponding parts are congruent. How? CPCTC. 3

Angle-Side-Angle

Triangle Congruence Postulate (ASA)

If two angles and the included side of one triangle are congruent to two angles and the included side of second triangle, then the two triangles are congruent.

A

C

B

D

F

E

4

Angle-Angle-Side or Side-Angle-Angle

Triangle Congruence Theorem (AAS or SAA)

If two angles and a non-included side of one triangle are congruent to the corresponding two angles and side of a second triangle, then the two triangles are congruent.

A

C

B

D

F

E

Do you see the difference between this and the sketch on page 4?

5

Hypotenuse – Leg

Triangle Congruence Theorem (HL)

If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of a second right triangle, then the two triangles are congruent.

C

BA

Notice how this is the exception or special case where SSA actually works. If the triangle is NOT a right triangle, you get the “Donkey Theorem.” (See page 7) 6

Side-Side-Angle or Angle-Side-Side (SSA or ASS)

also known as the “Donkey Theorem”

It is NOT a valid method to prove triangles are congruent to say that since two sides and a non-included angle of each are congruent that the triangles are congruent. It ain’t necessarily so! Check it out:

C

BA

BA

C

7

GRAPHIC ORGANIZER:

REMEMBER YOUR VOWELS

A + E + I + O = U AND Y

When submitting math work, always ask yourself:

A. Did I clearly identify and label or circle an ANSWER that makes sense for what I did?

E. Is my reasoning clearly and efficiently EXPLAINED?

I. Does my work reflect all the necessary INFORMATION I used to solve the problem?

Can others tell what letter, number and symbol represents?

O. Is my work well ORGANIZED with steps that follow a logical sequence?

U. If so, my work demonstrates that I UNDERSTAND the concepts involved.

… and always…

Y. Ask WHY.

Why did I select the strategies I did?

Why might others argue the benefits of a different strategy?

Why is the concept significant? …

GRAPHIC ORGANIZER:

RULES FOR PRESENTING

1. COME PREPARED.

2. VOLUNTEER TO PRESENT.

3. ADDRESS YOUR CLASSMATES.

4. SHOW RESPECT.

5. HELP FIND ERRORS.

6. CONTRIBUTE YOUR IDEAS.

7. POINT OUT CONNECTIONS.

8. CITE YOUR SOURCES.

GRAPHIC ORGANIZER:

PROBLEM SOLVING STRATEGY:

Identify the relationship. In math, how is what

is given and what you are asked to solve related?

Devise a plan. In math, write an equation.

Execute the plan. In math, solve the equation.

Answer the question that is asked. In math, this

may or may not be the solution to your

equation. Look at the question again.

GRAPHIC ORGANIZER:

STEPS FOR PROOF (shorthand used to explain your reasoning)

1. Write the GIVEN and PROVE statements.

(Identify and state the given information and what it is you are trying to prove).

2. Draw a SKETCH.

3. MARK the drawing. (the most important step)

A. First, mark only what is explicitly GIVEN.

B. Secondly, mark what is implied or implicitly true.

4. Set up a STATEMENT and REASON table.

5. Fill in the table. Justify each statement with a reason or rule.

SIX RULE CATEGORIES (used in this course)

1. Given Information

2. Definitions

3. Properties

4. Postulates

5. Theorems

6. Corollaries

IMPLICIT INFORMATION to watch for

• Vertical Pairs

• Common Side or Common Angle (Reflexive Property)

• Parallel Lines (and angle relationships formed by a transversal crossing parallel lines).

• Straight Angle

• Triangle Angle Sum

GRAPHIC ORGANIZER:

ENGLISH VOLUME CONVERSION

GRAPHIC ORGANIZER:

WRITE

SYMBOLS

A

l

AB

! "##

CD

AB

!ABC

m!ABC

AB

! "!!

!EFG

AB ! CD

AB ||CD

EF

! "##! GH

! "##

ABC

SAY

WORDS

Point A

Line L

Line AB

Segment CD

The measure of segment AB

Angle ABC

The measure of angle ABC

Ray AB

Triangle EFG

Segment AB is congruent to

segment CD

Segment AB is parallel to

segment CD

Segment EF is perpendicular

to segment GH

Parallelogram ABCD

Circle M

Plane ABC

DRAW

FIGURES

…

…

…

…

…

MODEL REAL WORLD

OBJECTS

Tip of a pen

Pointer

Intersection of ceiling and wall

Pencil

The length of the pencil

Bridge supports

The degrees of an angle

Pointer

Hill

Two people’s height

Rows of corn

Stairs and the railing

Street layout

Gear

White Board

GRAPHIC ORGANIZER:

TRANSFORMATIONS

GRAPHIC ORGANIZER:

BASIC TRIGONOMETRIC FUNCTIONS (SO-CA-TOA)

T o a

GRAPHIC ORGANIZER:

EXPONENTS

RULE EXAMPLE

GRAPHIC ORGANIZER:

CIRCLES

GRAPHIC ORGANIZER:

THE CARTESIAN COORDINATE PLANE

GRAPHIC ORGANIZER:

PROOF

GRAPHIC ORGANIZER:

QUADRATIC EQUATIONS

GRAPHIC ORGANIZER:

MULTIPLE REPRESENTATIONS – NUMERICAL, ALGEBRAIC, GRAPHICAL (NAG)

GRAPHIC ORGANIZER:

PROPORTION

GRAPHIC ORGANIZER:

SIMPLIFYING RADICALS

GRAPHIC ORGANIZER:

SIMILAR TRIANGLESGRAPHIC ORGANIZER:

SOLVING LINEAR SYSTEMS

Methods to Solve Systems of

LINEAR SYSTEMS

SUBSTITUTION

GRAPHING

COMBINATIONS

1. Graph Line 1.

2. Graph Line 2.

3. Visually identify the

point of intersection.

Intersecting

lines have

exactly one

solution

which is the

point of

intersection

(x, y)

Coinciding lines

have an infinite

number of

solutions (all the

points on the line

are solutions of

the system)

Parallel

lines have

no solution

because

there is no

point of

intersection

1. Isolate one variable in one of the

equations.

2. Substitute that expression into

the second equation and solve for

the variable.

3. Substitute the value found in

step 2 into either of the original

equations and solve for the

remaining variable.

4. Write the ordered pair.

1. Rewrite each equation in

standard form.

2. Choose a variable to eliminate

and multiply by appropriate

number to eliminate it.

3. Solve for remaining variable

either by substituting into one of

the original equations or by

repeating step 2 for the other

variable.

4. Write the ordered pair.

What if both of the variables cancel out? Look at the resulting arithmetic equation.

*False statement indicates the lines are parallel so there is no solution.

*True statement indicates the lines coincide so there are infinite solutions.

2x + 3y = 5

4x – y = 17