FIRST SIX WEEKS REVIEW. SYMBOLS & TERMS A B 6 SEGMENT Endpoints A and B

FIRST SIX WEEKS REVIEW

SYMBOLS & TERMS

A

B6

SEGMENT AB6AB

Endpoints A and B

A

BM

M is the Midpoint of ABAM MB

33

A segment has endpoints on a number line of -3 and 5, Find its length.

5 ( 3) 8

A segment has endpoints on a number line of -3 and 5, find its midpoint.

3 5 21

2 2

1 2 1 2,2 2

x x y y

1 2 1 2,2 2

x x y y

The Midpoint of a segmentThe Midpoint of a segment

Find the Find the midpointmidpoint of the of the segment joining (3,4) and segment joining (3,4) and (-5,-6).(-5,-6).

Find the midpoint of the Find the midpoint of the segment joining (segment joining (33,4) and ,4) and ((-5-5,-6).,-6).

1 2 1 2,2 2

yx x y

Find the midpoint of the Find the midpoint of the segment joining (segment joining (33,4) and ,4) and ((-5-5,-6).,-6).

1 23

2

5,

2

y y

Find the midpoint of the Find the midpoint of the segment joining (3,segment joining (3,44) and ) and (-5(-5,-6,-6).).

1 23 5,

2 2

y y

Find the midpoint of the Find the midpoint of the segment joining (3,segment joining (3,44) and ) and (-5(-5,-6,-6).).

3

2

65,

2

4

Find the midpoint of the Find the midpoint of the segment joining (3,4) and segment joining (3,4) and (-5,-6).(-5,-6).

3 5 4 6 2 2, ,

2 2 2 2

( 1, 1)

2 2 2a b c

2 2 2a b c Pythagorean Theorem– Pythagorean Theorem– Used to find a missing side Used to find a missing side of a right triangle.of a right triangle.

2 2 2a b c If a=5, and b=12, then c=_?_If a=5, and b=12, then c=_?_

13

2 2

2 1 2 1d x x y y

2 2

2 1 2 1d x x y y The distance formula—for finding The distance formula—for finding the length of a segment.the length of a segment.

Find the Find the distancedistance between between (-2,-6) and (4, 2): (-2,-6) and (4, 2):

Find the Find the distancedistance between between ((22,-6) and (,-6) and (-4-4, 2): , 2):

2

1

2

2 2 1d x x y y

Find the Find the distancedistance between between ((22,-6) and (,-6) and (-4-4, 2): , 2):

2 2

2 14 2d y y

Find the Find the distancedistance between between (2(2,-6,-6) and (-4, ) and (-4, 22): ):

2

2 1

24 2d y y

Find the distance between Find the distance between (2(2,-6,-6) and (-4, ) and (-4, 22): ):

2 24 2 2 6d

Find the Find the distancedistance between between (2,-6) and (-4, 2): (2,-6) and (-4, 2):

2 2

2 2

4 2 2 6

( 6) (8)

d

Find the distance between Find the distance between (2,-6) and (-4, 2): (2,-6) and (-4, 2):

2 2

2 2

4 2 2 6

( 6) (8) 36 64

d


2 2

2 2

4 2 2 6

( 6) (8) 36 64

100

d


2 2

2 2

4 2 2 6

( 6) (8) 36 64

10100

d

ACUTE AngleACUTE Angle

Less than 90

OBTUSE AngleOBTUSE AngleGreater than 90 but less than 180

RIGHT AngleRIGHT Angle

Equals 90

STRAIGHT STRAIGHT AngleAngleEquals 180

SPECIAL SPECIAL PAIRS OF PAIRS OF ANGLESANGLES

Nonadjacent Nonadjacent AnglesAngles

21

1

2

m ABC m CBD m ABD

B

C

A

D

For adjacent angles

______ ________ _______

m DAB m BAC m DAC

B

CA

D

______ ________ _______

m CBD m DBA m CBA

B

C

A

D

______ ________ _______

Supplementary Supplementary AnglesAngles

122 58

180m A m B A B

Vertical Vertical AnglesAngles

1

2

Also Vertical Also Vertical AnglesAngles

1 2

Linear PairLinear Pair

1 2

Complementary Complementary AnglesAngles 32

58A

B90m A m B

Congruent Congruent AnglesAngles

32

32A B

A

B

Angle BisectorAngle Bisector

21

1 2 B

C

Conditional Statement:

Any statement that is or can be written in if-then form. That is,

If p then q.

Symbolically we use the following for the conditional statement: “If p then q”:p q

EXAMPLE:

If you feed the dog, then you may go to the movies.

EXAMPLE:


Hypothesis

EXAMPLE:


Hypothesis

Conclusion

“ALL” Statements:When changing an “all” statement to if-then form, the hypothesis must be made singular.

EXAMPLE: All rectangles have four sides.BECOMES: If _______ a rectangle then _____ four sides.

a figure is it

has

The Converse:The conditional

statement formed by interchanging the hypothesis and conclusion.

Symbolically, for the conditional statement:p qThe converse is:

q p

EXAMPLE: Form the converse of:

If then

X=2

X > 0

.

EXAMPLE: Form the converse of:

If then

X=2

X > 0

.If then

X > 0

X=2

.

The Inverse:The conditional statement formed by negating both the hypothesis and conclusion.

Symbolically, for the conditional statement:p qThe inverse is:

p q

EXAMPLE: Form the Inverse of:

If then

X=2

X > 0

.If then

X=2

X > 0

.

EXAMPLE: Form the Inverse of:

If then

X=2

X > 0

.If then

X=2

X > 0

.

The Contrapositive:The conditional statement formed by interchanging and negating the hypothesis and conclusion.

Symbolically, for the conditional statement:p qThe contrapositive is: q p

EXAMPLE: Form the contrapositive of:If the

nX=2

X > 0

.If then

X=2

X > 0

.

LOGIC:SYLLOGISMS

Law of Syllogism

_________

p q

q r

p r

• If a figure is a rectangle, then it is a parallelogram.

• If a figure is a parallelogram, then its diagonals bisect each other.

• __________________________



• __________________________



• __________________________


• If a figure is a parallelogram, then its diagonals bisect.

• __________________________

If a figure is a rectangle, then its diagonals bisect.

Law of Detachment

________

p q

p

q


• ABCD is a rectangle.

• __________________________



• __________________________



• __________________________

ABCD is a parallelogram.

Law of Contrapositive

________

p q

q

p


• ABCD is not a parallelogram.

• __________________________



• __________________________



• __________________________

ABCD is not a rectangle.

In the following examples, use a law to draw the correct conclusion from the set of premises.

1. If frogs fly then toads talk.

Frogs fly.

-----------------------------

1. If frogs fly then toads talk.

Frogs fly.

-----------------------------

Toads talk.

2. If hens heckle then crows don’t care.

Crows care.

-----------------------

2. If hens heckle then crows don’t care.

Crows care.

-----------------------

Hens don’t heckle.

3. If ants don’t ask then flies don’t fret.

Ants don’t ask.

----------------------------

3. If ants don’t ask then flies don’t fret.

Ants don’t ask.

----------------------------

Flies don’t fret.

PROPERTIES

IF

then

AB BC

BC AB

Symmetric Property of Congruence

A A

Reflexive Property of Congruence

IF

and

then

AB BC

BC CD

AB CD

Transitive Property of Congruence

180m A m B If

and90m B 90 180m A then

Substitution Property of

Equality

IF

AB = CD

Then

AB + BC = BC + CD

Addition Property of Equality

If AB + BC= CE

and CE = CD + DE

then

AB + BC = CD + DE

Transitive Property of Equality

If

AC = BD

then

BD = AC.

Symmetric Property of

Equality

If AB + AB = AC

then 2AB = AC.

Distributive Property

m B m B

Reflexive Property of Equality

If

2(AM)= 14

then

AM=7

Division Property of Equality

If

AB + BC = BC + CD

then

AB = CD.

Subtraction Property of

Equality

If

AB = 4

then

2(AB) = 8

Multiplication Property of

Equality

Let’s see if you remember a few oldies but goodies...

If B is a point between A and C, then

AB + BC = AC

The Segment Addition Postulate

If Y is a point in the interior of

then

RST

m RSY m YST m RST

Angle Addition Postulate

IF M is the Midpoint

of

then

AB

AM MB

The Definition of Midpoint

IF

bisects

then

AB66666666666666

CADCAB BAD

The Definition of an Angle Bisector

If AB = CD

then

AB CD

The Definition of Congruence

If

then

is a right angle.

90m AA

The Definition of Right Angle

1

If

is a right angle, then the lines are perpendicular.

1

The Definition of Perpendicular

lines.

If

Then

A B

m A m B

The Definition of Congruence

And now a few new ones...

If and are right angles,

then

A B

A B

Theorem: All Right angles are congruent.

1 2

If and are congruent, then lines m and n are perpendicular.

n

m

1 2

Theorem: If 2 lines intersect to form congruent adjacent angles, then the lines are perpendicular.

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FIRST SIX WEEKS REVIEW. SYMBOLS & TERMS A B 6 SEGMENT Endpoints A and B