Math 2 Unit 2 Notes: DAY 1 – Review Properties & Algebra Proofs

Warm-up

Addition Property of equality (add prop =)

If a = b Then

If 5x-7 = 23 Then

If AB = CD Then AB+GH =

Subtraction Property of equality

If a = b Then

If 6y + 5 = -25 Then

If EF + AB = CD + AB Then

Multiplication Property of equality

If a = b Then

If m = 3 Then 4m = ____

If (½)mABC = 45 Then

Division Property of equality

If a = b Then

If 15y = 105 Then

If (5)EFG = 50 Then

Symmetric Property of equality

If a = b Then

If 5 = 3x – 1 Then

If -30 < x Then

Substitution Property of equality

If a = b and a+3=c Then

If 5x + 3 = z and x = -2 Then

If cat+3=dog & cat=n Then

Transitive Property of equality

If a=b and b=c Then

If 4+1=2+3 and 2+3=5 Then

If AB=CD and CD=EF Then

Reflexive Property of equality

a = a

AB = ______

mABC = _________

Distributive Property of equality

a(b+c) = ____________

4(x-5) = ____________

Definition of Congruence

If AB = CD Then

If mABC = mDEF Then

Notes

A proof is a _________________ argument in which each ________________ is

supported by a ________________ reason.

* This could be a _____________________, ____________________,

_________________________, etc.

There are 4 essential parts of a good proof:

1.

2.

3.

4.

When writing a reason for a step, you must use one of the following:

There are a few allowed assumptions:

Vertical Angles :_______________________________

Reflexive Property:_____________________________

Linear Pair:____________________________________

Any proof should start with the following:

Steps to write a proof:

1)

2)

3)

4)

EX 1: Solve 5x – 18 = 3x + 2 and write a reason for each step.

Statement Reason

5x – 18 = 3x + 2 given

EX 2: Solve 55z – 3(9z + 12) = - 64 and write a reason for each step.

Statement Reason

55z – 3(9z + 12) = - 64 given

Ex) Write the proof. Given: 𝑥

3+ 4 = 1 Prove: 𝑥 = −9

Math 2 Unit 3 Notes: DAY 2 – Review of Parallel Line

Warm-up

Finding the Slope of a Line

m =

m =

Give an example of an equation with a positive slope: __________________________

Give an example of an equation with a negative slope: ___________________________

Give an equation for a vertical line: ___________________________________

Give an equation for a horizontal line: _________________________________

Day 2 Notes

Transversal-

______________________________________________________________

______________________________________________________________

Draw a picture of a Transversal in the box.

_____________________________________--Angles on opposite sides of a

transversal and inside two other lines.

Draw a picture.

*__________________________________- If a transversal intersects two parallel

lines, then alternate interior angles are _________________.

_______________________________-Angles in the same position relative to a

transversal and two other lines

Draw a picture.

* __________________________________ – If a transversal intersects two parallel

lines, then corresponding angles are ____________________.

___________________________-Angles on the same side of a transversal and inside

two other lines

Draw a picture.

* ______________________________________ – If a transversal intersects two

parallel lines, then same-side interior angles are _____________________________.

____________________________________-Angles on opposite sides of a

transversal and outside two other lines

Draw a picture.

* ______________________________________ – If a transversal intersects two

parallel lines, then alternate exterior angles are _________________________.

Ex)

Use the diagram above. Identify which angle

forms a pair of same-side interior angles with

1. Identify which angle forms a pair of

corresponding angles with 1.

____________________________________-a pair of non-adjacent angles formed

when two lines intersect.

Draw a picture.

* ______________________________________ – If two lines intersect then

opposite angles are _________________________.

__________________________________-A pair of adjacent angles that form a line.

Draw a picture.

* ______________________________________ – A pair of adjacent angles

are _________________________.

For examples below, the figures shows p || q .

Ex) m 1 = x – 5 and m2 = 2x - 40, find x and m 1.

x = ______

m 1 = ______

Ex) m3 = 6x + 12 and m 4 = 10x + 8, find x and m 4.

x = ______

m4= ______

q

t

1

2

p

q

t

3

4

p

We can use a transversal to prove lines parallel and relate parallel and perpendicular lines.

We do that using the _____________________ of the parallel lines theorems.

Ex)

Ex)

Ex)

Math 2 Unit 3 Notes: DAY 3 – Parallel Line Proofs

Warm-up

Assume a ⁄⁄ b. Complete the chart.

ANGLES TRANSVERSAL TYPE , SUPPLEMENTARY, OR NONE

1. 1 and 14

2. 2 and 15

3. 7 and 9

4. 9 and 16

5. 10 and 17

6. 16 and 14

7. 9 and 14

8. 18 and 19

9. 1 and 16

10. 3 and 8

11. 6 and 9

2

5

1 6

3 4

9

11 13 15

b 7

8

10 12

14

16

17

18 19

x

y a

Notes

Writing a parallel line proof is similar to writing an algebraic proof, the only difference is you use

the ________________________ ____________ ___________________ to justify your

reasons.

Can you remember all the properties? List them below:

1)

2)

3)

4)

5)

6)

7)

If you want to prove that lines are parallel from these properties remember to use the

_______________________.

Ex) Prove the following:

a) Given: l // m; s // t

Prove: 1 5

__________________________________________________________________________________

1. l // m; s // t 1. _________________________________________

2. 1 3 2. _________________________________________

3. 3 5 3. _________________________________________

4. 1 5 4. _________________________________________

b) Given: l // m; s // t

Prove: 2 4

__________________________________________________________________________________

1. l // m ; s // t 1. ________________________________________

2. 2 3 2. ________________________________________

3. 3 4 3. ________________________________________

4. 2 4 4. ________________________________________

c) Given: l // m; 1 4

Prove: s // t

__________________________________________________________________________________

1. l // m ; 1 4 1. _______________________________________

2. 3 1 2. _______________________________________

3. 3 4 3. _______________________________________

4. s // t 4. _______________________________________

d) Given: l // m ; 2 5

Prove: s // t

__________________________________________________________________________________

1. l // m ; 2 5 1. ________________________________________

2. 2 3 2. ________________________________________

3. 3 5 3. ________________________________________

4. s // t 4. ________________________________________

Ex. Given: l // m ; s // t

Prove: 2 4

Ex. Given: l // m ; s // t

Prove: 1 5

l

m

s t

1

2

3 5

4

Math 2 Unit 3 Notes: DAY 4 – Congruent Triangles

Warm-up

Definitions and Postulates Regarding Segments

Segment Addition Postulate

If C is between A and B,

Then_____________________

Definition of Segment Congruence

If 𝐴𝐵̅̅ ̅̅ ≅ 𝐶𝐷̅̅ ̅̅ ,

Then ___________________

Definition of Segment Bisector

If 𝐴𝐵̅̅ ̅̅ 𝑏𝑖𝑠𝑒𝑐𝑡𝑠 𝐶𝐷̅̅ ̅̅ ,

Then ____________________

Definition of Midpoint

If B is the midpoint of A and C,

Then ______________________

Research the properties above and finish the statement. Then illustrate the Definition

or Postulate below.

Segment Addition Postulate Definition of Segment Congruence

Definition of Segment Bisector Definition of Midpoint

Notes

Congruent Triangles: triangles that are the same ________ and ________

Each triangle has three ___________ and three _________.

If all pairs of the corresponding parts of two triangles are __________________,

then the triangles are ________________.

Congruent Triangles:

Corresponding Congruent Angles:

Corresponding Congruent Sides:

Example #1: In the following figure, QR = 12, RS = 23, QS = 24, RT = 12,

TV = 24, and RV = 23.

Name the corresponding congruent

angles and sides.

Properties of Triangle Congruence:

Reflexive Symmetric Transitive

Example #2: If STJWXZ , name the congruent angles and congruent sides.

Angles –

Sides –

Naming Congruent Triangles

I. Draw and label a diagram. Then solve for the variable and the missing

measure or length.

1. If ∆𝐵𝐴𝑇 ≅ ∆𝐷𝑂𝐺, and 𝑚∠𝐵 = 14, 𝑚∠𝐺 = 29, 𝑎𝑛𝑑 𝑚∠𝑂 = 10𝑥 + 7. Find the value of x

𝑚∠𝑂.

x = ___________

𝑚∠𝑂= _________

2. If ∆𝐶𝑂𝑊 ≅ ∆𝑃𝐼𝐺, and 𝐶𝑂 = 25, 𝐶𝑊 = 18, 𝐼𝐺 = 23, 𝑎𝑛𝑑 𝑃𝐺 = 7𝑥 − 17 . Find the value of

x and PG.

x = ___________

PG=___________

3. If ∆𝐷𝐸𝐹 ≅ ∆𝑃𝑄𝑅 and 𝐷𝐸 = 3𝑥 − 10, 𝑄𝑅 = 4𝑥 − 23, 𝑎𝑛𝑑 𝑃𝑄 = 2𝑥 + 7. Find the value of x

and EF.

x = ___________

EF = __________

II. Use the given information and triangle congruence statement to complete the

following.

1. ∆𝐴𝐵𝐶 ≅ ∆𝐺𝐸𝑂, AB = 4, BC = 6, and AC = 8.

2. What is the length of 𝐺𝑂̅̅ ̅̅ ? How do you know?

3. ∆𝐵𝐴𝐷 ≅ ∆𝐿𝑈𝐾, 𝑚∠𝐷 = 52°, 𝑚∠𝐵 = 48°, 𝑎𝑛𝑑 𝑚∠𝐴 = 80°.

a. What is the largest angle of ∆𝐿𝑈𝐾?

b. What is the smallest angle of ∆𝐿𝑈𝐾?

Side–Side–Side Congruence: If the ___________ of one triangle are congruent to the

sides of a second triangle, then the triangles are ___________________.

Abbreviation:

Side–Angle–Side Congruence: If two sides and the included ____________ of one

triangle are congruent to two ___________ and the included angle of another triangle,

then the triangles are __________________.

Abbreviation:

Example #1: Mark the figure & state if the triangle is congruent by SSS or SAS.

Given: HIFE , and G is the midpoint of both EI and FH .

Example #2: Mark the figure & state if the triangle is congruent by SSS or SAS.

Given: DE and BC bisect each other.

Example #3: Mark the figure & state if the triangle is congruent by SSS or SAS.

Given: ACAB and CYBY

Math 2 Unit 3 Notes: DAY 5 – ASA, AAS & HL

Warm-up

Write a 2-column proof.

1) Given: //k l

Prove: 6 is supp. to 7 .

2) Given: //k l

Prove: 2 7

k

l

t

3 7

84

5

6

1

2

Notes

Angle–Side–Angle Congruence: If two _____________

and the included _________ of one triangle are

congruent to two angles and the included side of another

triangle, then the triangles are

_____________________.

Abbreviation:

Angle–Angle–Side Congruence: If two angles and a non-

included side of one triangle are congruent to the

corresponding two ______________ and a side of a second

triangle, then the two triangles are

____________________.

Abbreviation:

Hypotenuse Leg Coungruence: If one angle measures 90 degrees and both have a

congruent leg and hypotenuse then both triangles are congruent.

Abbreviation:

Example #1: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL.

Given: AB bisects CAD and 1 2

Example #2: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL.

Given: AD CB and A C

Example #3: Mark the figure and state if the triangle is congruent by ASA, AAS, or HL.

Given: V S and TV QS

After Quiz Practice: Triangle Congruence (SSS-SAS-ASA-HL-AAS)

SSS: (Side-Side-Side)

If the sides of one triangle are

congruent to the sides of

another triangle, then the two

triangles are congruent.

ABC DEF

SAS: (Side-Angle-Side)

If two sides and the included angle in

one triangle are congruent to two sides

and the included angle in another

triangle, then the two triangles are

congruent.

ABC DEF

ASA: (Angle-Side-Angle)

If two angles and the included side in

one triangle are congruent to two angles

and the included side in another triangle,

then the two triangles are congruent.

ABC DEF

AAS: (Angle-Angle-Side)

If two angles and a nonincluded side of one triangle

are congruent to the corresponding angles and

nonincluded side of another triangle, then the two

triangles are congruent.

ABC DEF

HL: (Hypotenuse-Leg)

If the hypotenuse and a leg of one right triangle are congruent

to the hypotenuse and corresponding leg of another right

triangle, then the two triangles are congruent.

ABC DEF

Determine whether each pair of triangles can be proven congruent by using the SSS, SAS, ASA, HL or AAS

Congruence Postulates. If so, write a congruence statement and identify which postulate is used. If not, write

“cannot be proven congruent.”

1)

2)

3)

4)

5)

6)

7)

8)

9)

10)

11)

12)

Math 2 Unit 3 Notes: DAY 6 – Triangle Congruency Proofs

Warm-up

For each pair of triangles, tell which conjectures, if any, make the triangles congruent.

1. ABC EFD ______________ 2. ABC CDA ______________

3. ABC EFD ______________ 4. ADC BDC ______________

5. MAD MBC ______________ 6. ABE CDE ______________

7. ACB ADB ______________ 8. ______________

C

A B D

F

E

A

C

B D

F

E A B D

C

C

A D

B

A

C

D

B

D

A

C

B M A B

E

C D

DA C

B

Notes

Recall the steps for writing a proof:

1)

2)

3)

4)

The only difference in writing a proof for Congruent Triangles is the final

statement. The final statement should include one of the following: __________,

___________, ____________, _____________, and ____________.

Ex 1) Given: 𝐴𝐷̅̅ ̅̅ ≅ 𝐷𝐶̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ ⊥ 𝐵𝐷̅̅ ̅̅ Prove: ΔABD ≅ ΔCBD

DA C

B

Ex 2) Given: <E ≅ <H G is the midpoint of 𝐸𝐻̅̅ ̅̅ Prove: ΔGFE ≅ ΔGIH

Ex 3) Given: 𝐴𝐵̅̅ ̅̅ ≅ 𝐵𝐶̅̅ ̅̅ 𝐴𝐶̅̅ ̅̅ ⊥ 𝐵𝐷̅̅ ̅̅ Prove: ΔABD ≅ ΔCBD

G

F

E

H

I

SCRAMBLE PROOFS: Draw the figure and write final proof below.

Math 2 Unit 3 Notes: DAY 7 – Triangle Congruency Proofs

Warm-up

1. write a congruency statement for the two triangles at right.

2. List ALL of the congruent parts if EFG HGF

3. Name all the ways to prove triangles congruent.

For each pair of triangles, tell: (a) Are they congruent (b) Write the triangle

congruency statement. (c) Give the conjecture that makes them congruent.

4.

5.

C

A R

G

E

O

A

B

C

D

A

W

T

E

R

Practice: Fill in the following proofs with the necessary Statements and Reasons to

prove the triangles congruent. 1)

2)

Statements Reason

Statements Reason

3) Given: O is the midpoint of MQ O is the midpoint of NP Prove:

Statements Reasons

4)

Statements Reason

5)

Given: AD || EC

BD BC

Prove: ∆ ABD ∆ EBC

Statements Reasons

Math 2 Unit 3 Notes: DAY 8 – CPCTC

Warm-up

Given: B C

D F

M is the midpoint of DF.

Prove: ∆ BDM ∆ CFM

Statements Reasons

Notes

CPCTC- Corresponding Parts of Congruent Triangles are Congruent

*Explanation: To prove that parts (sides or angles) of triangles are congruent to parts of other

triangles, first prove the triangles are congruent.

Then by CPCTC, all other corresponding parts will be congruent.

Statements Reasons

Ex) Given: AB DC ;

AD BC

Prove: A C

When writing a proof, ___________ should be your ____________ reason!!!!!

Ex) Given: MA TA , A is the midpoint of SR

Prove: MS TR

Ex) Given: 1 2 ; 3 4

Prove: CB CD

Statements Reasons

Statements Reasons

Ex) Given: MS || TR; MS TR

Prove: MA TA.

Statements Reasons