Reasoning and Proof Algebra Properties of Equalityteachers.sduhsd.net/mchaker/Honors...

Reasoning and Proof

Algebra Properties of Equality

If a b and x y, then a x b y.

If a b and x y, then ax by.

. ,segment someGiven ABABAB

. , ngle someGiven AAAa

If a b and b c, then a c.

Addition

(APE)

Multiplication

(MPE)

Reflexive

Substitution

l m

x y

z

Given: Intersecting lines l and m with two

vertical angles having measures x and y.

Prove: x = y

Statements Reasons

1. x + z = 1800

2. y + z = 1800

3. x + z = y + z

4. x = y

1. Definition of a straight

angle

2.

3.

4.

Definition of a straight

angle

Substitution (steps 1 & 2)

APE

Statements Reasons

1.

2.

3.

4.

5.

6.

7.

8.

9.

Prove the following conditional:

If two angles are congruent and supplementary, then they

are right angles.

anglesright are B andA :Prove

.arysupplement are B ndA and :Given

aBA A B

BmAm oBmAm 180oAmAm 180oAm 1802

oAm 90oBm 90

anglesright are B andA

Def. of supplementary angles


Simplification

MPE


Definition of a right angle

Definition of congruent angles

m A m B

A and B are supplementary Given

Given

Congruent Triangles

Two figures are CONGRUENT if they have

exactly the same size and shape.

When two figures are congruent, we always list

CORRESPONDING LABELS in the same order.

Triangles have 6 parts (3 angles and 3 sides). When

triangles are congruent, their corresponding parts are also

congruent. List the 6 pairs of corresponding parts from the

congruent triangles above. DEAB

DFAC

EFBC

EDFBAC

FEDCBA

EFDBCD

Congruent Triangles

When all six parts of one triangle are

congruent to the corresponding 6 parts of

another, then the two triangles are congruent.

For example, would knowing that two pairs of sides are

congruent be enough?

However, what is the minimum amount of information

required to conclude that two triangles are congruent?

Key Concept:

If the information you provide is sufficient, then you should

be able to create only one unique triangle with the given

conditions.

Use the straws provided to represent sides of triangles and

determine what the minimum conditions are to conclude that

two triangles are congruent. Make a list of these conditions.

Congruent Triangles

Why doesn’t Angle, Side, Side work?

Reason 1:

ASS is a BAD word!!

A

S

S S

Also, there is a counterexample.

The triangle below is not the only one possible with

the given conditions.

Congruent Triangles

Why does AAS work?

A

B

C D

E

F

o25

o25

o75o75

8

8

What do you know about angles C and F?

o80 o80

So what other triangle congruence theorem

does AAS come from?

Congruent Triangles

However, ASS does work when the

Hypotenuse and one Leg of a Right

Triangle are congruent. Why?

C

A

C

A

B B

The Pythagorean Theorem forces the third pair of sides

to be congruent, giving us SAS. This Triangle

Congruence Postulate Is called Hypotenuse-Leg (HL).

Congruent Triangles

What if the legs of two right triangles

are congruent? Are the triangles

congruent?

A A

B B

This Triangle Congruence Postulate Is called Leg-Leg (LL).

It is derived from SAS.

Congruent Triangles

So, the Triangle Congruency Postulates are:

Side-Angle-Side (SAS)

Two congruent sides and their included angle.

Side-Side-Side (SSS)

All three sides are congruent.

Angle-Side-Angle (ASA)

Two angles and their included Side are congruent.

Angle-Angle-Side (AAS)

Two angles and a side that is NOT included by them.

These postulates represent the minimum amount of information

considered sufficient for two triangles to be congruent.

Congruent Triangles

The Right Triangle Congruency Postulates are:

Hypotenuse-Leg (HL)

If the hypoteni and one leg of two right triangles are congruent,

then the triangles are congruent.

In order to use these postulates as reasons for triangle congruency

in proofs, you would have to first demonstrate that the triangles are

right triangles.

Leg-Leg (LL)

If the legs of two right triangles are congruent, then the

triangles are congruent.

D

Definitions, Postulates, and Theorems

A B C

Segment Addition Postulate (SAP)

ACBCAB

B

A C

m BAD m DAC m BAC

Angle Addition Postulate (AAP) A B C

D

180om ABC

180om ABD m DBC

arysupplement are and DBCABD

Definition of a Straight Angle

D

Definitions, Postulates, and Theorems

A B C

B

A C

Definition of a Segment Bisector

Definition of an Angle Bisector

Altitude

A

B C D

Median

Equally Wet

Statements Reasons

1

2

3

4

5

6

7

8

Given:

Prove:

DCAB and,AB ofmidpoint theis C

BDAD

AB ofmidpoint theis C

CBAC

anglesright are BCD and ACD

BDAD

Given

CDAB Given

Midpoint a of Def

A C B

D

SAS

Theorem Congruency AngleRight

of Definition

Property ReflexiveCDCD

BCD ACD

BCD ACD

CP

Equally Wet

Statements Reasons

1

2

3

4

5

6

7

8

Given:

Prove:

DCAB and,AB ofmidpoint theis C

BDAD

AB ofmidpoint theis C

CBAC

anglesright are BCD and ACD

BDAD

Given

CDAB Given

Midpoint a of Def

A C B

D

LL

of Definition

Property ReflexiveCDCD

BCD ACD

CP

Equally Wet

nglesright tria are BCD and ACD ngleright tria a of Definition

Given:

Prove:

D ofbisector angle the and with DCBDADADB

ABDCCBAC and

BDCADC

BDAD

DCDC

BDCADC

CBAC

BCDACD

anglesright are B and CDACD

AB DC

Given

DDC bisects Given

bisector anglean ofn Defintitio

Property ReflexiveA C B

D

SAS

CP

CP

Thm. sSupplementCongruent

of Definition

anglestraight a is ACB Given

arysupplement are nd BCDaACD anglestraight a of Definition

Theorem 4.9

Statements Reasons

A C

B

The Base Angles Theorem

CBDABD

Isosceles triangles are symmetric. If we draw

in the angle bisector of B, the triangle is

symmetric about this axis of symmetry.

Since the triangle is symmetric:

CA CP

What is the Converse of

this Theorem?

Is it true?

The Base Angles Theorem

How could we use this theorem to explain why

all equilateral triangles are also equiangular?

A C

B What is the

Converse of

this Theorem?

Is It True?

Exterior Angle Theorem

a d c

b

Which groups of angles equal 180?

Can you use substitution to show that angle d is equal to the

sum of angles a and b?

ocba 180 odc 180

dccba

dba

The measure of an exterior angle of a triangle is equal to the

sum of the measures of the non adjacent interior angles (and

consequently greater than the measures of either).

_

From Two Flowers to Three

(4,2) (14,2)

(4,8)

_

| |

1

2

3

4

5

6

7

8

Given:

Prove:

AB to DCsuch that drawn is DC , DBAD

AB ofbisector theis DC

DBAD

DCDC

BDCADC

CBAC

Given

Property Reflexive

A C B

D

HL

CP

The Perpendicular Bisector Theorem Converse

ABDC

ABDC ofBisector theis Bisector a of Definition

Given

s'right are and BCDACD of Definition

sright are and BCDACD right of Definition

1

2

3

4

5

6

7

8

9

Given:

Prove:

AB to DCsuch that drawn is DC , DBAD


DBAD

CBAC

Given

A C B

D

CP


ABDC


Given

s'right are and BCDACD of Definition

BCDACD Theorem Right

Isosceles is ABD Triangles Isosceles of Def

CBDCAD Theorem Angles Base

CBDCAD AAS

1

2

3

4

5

6

7

8

Given:

Prove: AB ofmidpoint theis Cuch that drawn is DC

s

DBAD


DBAD

DCDC

BDCADC

Isosceles is ADB

BDCADC

Given

midpoint a of Definition

Property Reflexive

A C B

D

SSS

Triangles Isosceles of Def

CP


CBAC

4.9 Theorem

ADB bisects DC bisector of Def

ABDC ofBisector theis

1

2

3

4

5

6

7

8

9

10

Given:

Prove: AB ofmidpoint theis Cuch that drawn is DC

s

DBAD


DBAD

DCDC

BDCADC

supp. are and BCDACD

BCDACD

Given

midpoint a of Definition

Property Reflexive

A C B

D

SSS

AngleStraight a of Def

CP


CBAC

sSupplementCongruent

straight a is ACB Given

sBCDACD right are and

ABDC Definition


Reasoning and Proof Algebra Properties of Equalityteachers.sduhsd.net/mchaker/Honors...

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