1 34 Common Core Math 3 Notes - Unit 2 Day 1 Introduction ...rech-delany.weebly.com/uploads/2/4/0/8/24081973/... · CP Math 3 Page 8 of 34 Common Core Math 3 Notes - Unit 2 Day 3

CP Math 3 Page 1 of 34

Common Core Math 3

Notes - Unit 2 Day 1

Introduction to Proofs

Properties of Congruence

Reflexive A A

Symmetric If A B , then B A

Transitive If A B and B C

then A C

Properties of Equality

Reflexive A = A

Symmetric If A = B, then B = A

Transitive If A = B and B = C,

then A = C

Distributive a(b + c ) = ab + ac

Substitution If a + b = c and b = f,

Then a + f = c.

Addition If a = b, then a + c = b +c

Subtraction If a = b, then a – c = b – c

Multiplication If a = b, then 3a = 3b.

Division If a = b, then a/3 = b/3.


Proofs – YOU MUST SHOW ALL THE STEPS IN AN ALGEBRAIC PROOF!!!!

Examples:

1. Given: 3(x – 2) = 42

Prove: x = 16

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

2. Given: 7m + 3 = 6

4

Prove: m = 3

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.


Practice:

3. Given: -3(a + 3) + 5(3-a) = -50

Prove: a = ?

Statements: Reasons:

4. Given:

Prove: p = ?



5. Given:

Prove: a = ?



Common Core Math 3


Segments

What’s the difference???? Geometry: It’s all about the SYMBOLS and the PICTURES!!!

AB

AB AB

Example:

Line Segment – What do you notice about the segments?

●Definition of Equality: ●Definition of Congruence:

●Definition of Midpoint – midpoint of a segment is the point halfway between the endpoints of the segment

If X is the midpoint of AB , then AX = XB

●Midpoint Theorem – If X is the midpoint of AB , then AX XB

Ex: X is the midpoint of AB Hints:

Draw and label picture

AX = 3y + 7 Make sure you answer the question.

XB = 4y – 2

Find AB.

5units

M

A T H

5units


●Definition of Segment Bisector- any segment, line or plane that intersects a segment at its midpoint

To “bisect” a segment means to

Sketch each of the following; be sure to include appropriate “marks.”

CD bisects AB at x AB bisects CD at x AB and CD bisect each other at x

●Definition of BETWEEN - refers to points on a line, ray or segment

T is between S and U

1. Find x if T is between S and U and ST = 7x, SU = 45, TU = 5x – 45.

2. A is between B and C. BA = x2, AC = 6x + 10, and BC = 17. Find x and the length of each segment.

A

A A

B

B B

Draw & Label picture!


3. L is between K and M. KL = x2 – 10, LM = 5x + 4, and KM = 2x

2 – 42. Find x.

Segment Addition Postulate -

Ex 1: Given AC

AC = 3y + 1

AB = 2y

BC = 21

Find AB.

Find the value of the variable and LM if L is between points N and M.

Ex 2: NL = 5x, LM = 3x, NL = 15 Ex 3: NL = 5x – 3, NM = 2x + 6, LM = x – 7

D O G


Common Core Math 3


Angles

Definition of Equality: Definition of Congruence:

Definition of Linear Pair – a pair of adjacent angles whose non-common sides are opposite rays

Definition of Vertical Angles – two nonadjacent angles formed by two intersecting lines. Notice that the definition just describes the picture of the angles, not the relationship.

Vertical Angles Theorem - If two angles are vertical angles, then they are congruent. Notice that the theorem describes the relationship between the 2 angles.

Ways to name angles:

1.

2.

3.

4.

1 2


Angle Addition Postulate –

If A is on the interior of <DOG then, __________ + ____________ = ____________

So if you see…… You must be able to write…….

Angle Bisector 1. Label <CAT

2. Draw and label AB such that it bisects <CAT

3. Place the appropriate “marks” to indicate the bisector

4. Write an equality statement about the two angles.

CAREFUL!!! These are not the same! Solve for x in each.

1. PN bisects <MPR 2. <CAT = 10x 3. <CAT is a right angle

<MPN = 2x + 14 <CAD = 7x <CAD = 7x

<NPR = x + 34 <DAT = 15 <DAT = 15

O L

E

V

D D C C

A T

T A

Hmmmm…same picture……..as above!

L


Definition of Complementary Angles– Definition of Supplementary Angles–

2 angles whose sum is _________________

<A is comp to <B

2 angles whose sum is ________________

<J is supp to <D

Statement: Statement:

Ex: The supplement of an angle measures 78 degrees less than the measure of the angle. What are the

measures of the angle and the supplement?

4. 5.


Supplement Postulate – Complement Postulate

If two angles form a linear pair, then they are

supplementary

If the non-common sides of two adjacent angles for a

right angle, then the angles are complementary.

If I see this picture I can write……

If I see this picture I can write……

Supplementary Angles Theorem Complementary Angles Theorem Angles supplementary to the same angle Angles complementary to the same angle

or to congruent angles are congruent or to congruent angles are congruent.

Given: <A is sup to <C and A B Given: <D is comp to <Y and A D

Prove: <B is sup to <C Prove: <A is comp to <Y

Statement Reason Statement Reason

1 1. 1. 1.

2.

2. 2. 2.

3. 3. 3. 3.

4. 4. 4. 4.

5. 5. 5. 5.

6. 6. 6. 6.

A

B

C

D

A Y

non-common side

non

-com

mo

n sid

e


Definition of Perpendicular Lines: Lines that form right angles

Sketch: Symbol: Slopes:

Theorem: Perpendicular lines intersect to form 4 right angles

Theorem: All right angles are congruent.

If you know that <A is a right angle and <B is right angle, prove that <A is congruent to <B.

Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.


Common Core Math 3

Unit 2 Day 4

Segment Proofs!!!

Getting Ready for Proofs

For each of the following “givens” state the conjecture and the reason.

# Given Conjecture Reason

1. <A is complementary to <B

2. T is the midpoint of BE

3. T is the midpoint of BE

4. <A is supplementary to <B

5. ID= OL

6. ID = OL

7. ID = OL

OL = ME

8. m<1 = m<2

m<4 +m <2 = 180

9. <A is a right angle

10. AB is perpendicular to BC

11.

AB ≅ CD

12. C H S

13. B

U

G Y


N

M

P

Q

R S

14. TA is an angle bisector

M

A

T H

15.

1 2

16.

1 2

17.

1 2

18. S is between P an U

19. AB BC

20. <3 and <4 are vertical

PROOFS!!!

#1.

Given: MN PQ ; PQ RS

Prove: MN RS

Statements Reasons

1. MN PQ 1.

2. PQ RS 2.

3. MN RS 3.


E

D

A

C

B

PS

QR

#2.

Given: BC = DE

Prove: AB + DE = AC

Statements Reason

1. 1.

2. 2.

3. 3.

.

#3.

Given: Q is between P and R

R is between Q and S

PR = QS

Prove: PQ = RS

Statements Reason

1. 1.

2. 2.

3. 3.

4.

4.

5.

5.

6.

6.

7.

7.

.


PS

QR

A

BM

#4.

Given: PR QS

Prove: PQ RS

Statements Reason

1. 1.

2. 2.

3. 3.

4.

4.

5.

5.

6.

6.

#6.

Given: M is the midpoint of AB

Prove: 2AM = AB

Statements Reason

1. 1.

2. 2.

3. 3.

4.

4.

5.

5.


A

ED

B

C

Common Core Math 3

Unit 2 Day 5

Angle Proofs

Steps for parts of some proofs:

To prove two angles = 180⁰ To prove angles = 90⁰

1. Prove LP

2. Prove Supp

3. If Supp then = 180

1. Prove perpendicular

2. If perpendicular then right <

3. If right < then = 90⁰

1.

Given: AEC BED

Prove: AEB CED

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

2.

Given: AEB CED

Prove: AEC BED

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.


Proof of the Vertical Angle Theorem:

The Vertical Angle Theorem states: ________________________________________________

#3.

Given: 2 intersecting lines

Prove: <1 ≅ <3

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

10. 10.

#4.

Given: <1 and <3 are supplementary

<3 and <4 are linear pairs

Prove: <1 ≅ <4 4 3 2 1

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8

9. 9.

10. 10.

1 3 2


#5.

Given: <1 and <2 form a linear pair

<2 ≅ <1

Prove: <1 and <2 are right angles

Statements Reasons

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

9. 9.

10. 10.

11. 11.

12. 12.


Common Core Math 3

Unit 2 Day 6 Parallel Lines

Definition:

Sketch: Symbol: Slopes:

Skew Lines –

Transversal –

Angles formed by 2 lines and a transversal:

Alternate interior

Alternate exterior

Consecutive or same side interior

Corresponding

8 7 6 5

2 1 4 3


INVESTIGATION – You will need two different colored highlighters.

Investigation 1:

1. Trace with your highlighter. What letter do you see?

2. Name the typs of angle created?

3. What is the relationshp between those two angles?

Investigation 2:




Investigation 3:



9. What is the relationshp between those two angles


Investigation 4:




Investigation 5:




PRACTICE!!

1. Name alt int <s using line x as the transversal:

2. Name s.s. int <s using line y as the transversal:

3. Name corr <s using line z as the transversal:

4. Name alt ext <s using line y as the transversal:

5. Name alt int <s using line z as the transversal:

6. Name s.s. int <s using line z as the transversal:



3. Write an equation and solve for the unknown. State the theorem used to make the equation.

a)

b)

c)

d)

Use the diagrams to find x and y:

e. f.

:

g.


Common Core Math 3

Unit 2 Day 7 Proofs with Parallel Lines

Theorem: If given a line and a point not on the line, then there exists exactly one line through the point

that is parallel to the given line.

Example: Find the measure of the angle noted by the “ ? ”

Practice:

1. Solve the crook problem to find the missing angle:

2. Solve the crook problem to find the missing angle:

65˚

60˚

?


Using the theorems presented previously, prove that the sum of the degrees of the angles in a triangle

measure 180 degrees.

Given: Triangle CHS

Prove: m<C + m<H + m<S = 180o

Statements Reasons

1.

1.

2.

2.

3.

3.

4.

4.

5.

5.

6.

6.

Theorem – Perpendicular Transversal Theorem – In a plane, if a line is perpendicular to one of two

parallel lines, then it is perpendicular to the other.


m 1 2

3 4

6

7 8

5 p

z

m 1 2

3 4

6

7 8

5 p

z

1. Given: 1 5;

Prove:4 is supplementary to 6

m 1 2

3 4

6

7 8

5p

z

2. Given: 4 is supplementary to 6

Prove 3 7

3. Given:3 is supplementary to 8

Prove: 4 5


m 1

2

3

4

6

7

8

5

p

z

9

10

11

12

13

14

15

16

y

m 1

2

3

4

6

7

8

5

p

z

9

10

11

12

13

14

15

16

y

4. Given: 5 13, 10 15

Prove: 2 12

5. Given: 2 is supplementary to 3,

1 is supplementary to 13

Prove: 4 10


Common Core Math 3

Unit 2 Day 8

Proving Triangles Congruent

Side–Side–Side Congruence:

If the sides of one triangle are congruent to the sides

of a second triangle, then the triangles are

congruent.

Side–Angle–Side Congruence:

If two sides and the included angle of one

triangle are congruent to two sides and the

included angle of another triangle, then the

triangles are congruent.

Angle–Side–Angle Congruence: If two angles and the included side of one

triangle are congruent to two angles and

the included side of another triangle, then

the triangles are congruent.

Angle–Angle–Side Congruence:

If two angles and a non-included side of one

triangle are congruent to the corresponding

two angles and a side of a second triangle,

then the two triangles are congruent.

Hypotenuse-Leg Congruence:

If the hypotenuse and one leg of a right triangle are

congruent to the hypotenuse and the corresponding

leg of another triangle then the triangles are

congruent.

Abbreviation:

Not all triangles are congruent:

What about AAA?? What about SSA??

CP Unit 2 Notes Page 30

A

B

C

D

S

R

V

T

U

Proving “Triangles congruent”

Statements Reasons

1) 1) Given

2) angle or side stated 2)



5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL

_________________________________________________________________________________________________

Congruent Triangle Proof Examples

1. Given: ACD ACB

CD CB

Prove: ACD ACB

Statements Reasons

1.

ACD ACB

CD CB

1.

2. 2.

3. ACD ACB 3.

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

2. Given: RS US

ST SV

Prove: RSV UST

Statements Reasons

1.

RS US

ST SV

1.

2. 2.

3. RSV UST 3.

Picture/Diagram

required! Include tic

marks, arc marks, color,

“A” or “S” label.


M

L

P

N

Q O

3. Given: LP NP

NQP LOP

Prove: NQP LOP

Statements Reasons

1. LP NP

NQP LOP

1.

Redraw each triangle and label 2. 2.

3. NQP LOP 3.

Proving “Triangle PARTS”

C P C T C

Given:

Prove: part part

Statements Reasons

1) 1) Given




5) Δ Δ 5) SSS, SAS, ASA, AAS, or HL

6) part part 6) CPCTC


A E

GB

F

A

B C

D

Congruent “Triangle PARTS” Proof Examples

4. Given: || and ||BC AD AB CD

Prove: <B <D

Statements Reasons

1) 1)

2) 2)

3) 3)

4) 4)

5) 5)

6) 6)

5. Given:

AB FG

AB FG

B G

Prove: AF FE

Statements Reasons

1) 1)

2) 2)

3) 3)

4) 4)

5) 5)

6) 6)


G

F

O N

M

6. Given: FON MNO

FO MN

Prove: F M

Statements Reasons

1)

; FON MNO FO MN 1)

2) 2)

3) 3)

4) 4)

Isosceles Triangle Theorem - If two sides of a triangle are congruent, then the angles opposite

those sides are congruent

Statement Reason

1. 1.

2. 2.

3. 3.

4. 4.

5. 5.

6. 6.

7. 7.

8. 8.

Given: BCAC

Prove: BA

(hint: make CD an angle bisector)


Documents

1 34 Common Core Math 3 Notes - Unit 2 Day 1 Introduction ...rech-delany.weebly.com/uploads/2/4/0/8/24081973/... · CP Math 3 Page 8 of 34 Common Core Math 3 Notes - Unit 2 Day 3