Name :______________________ all dates are

subject to change

Day Topic Assignment

1 4.1 – Triangle Angle-Sum Theorem pg 221 # 14-28 even 32-34

4.2- Congruent Figures pg 228 #5-11,26

2 Quiz

4.3- 4.5 –Triangle Congruence pg 236 # 5-8, 9, 18,24, 26

SSS, SAS, ASA, AAS and HL pg 243 #3,4, 9-11, 20-22, 35,36

3 Continue previous lesson with more

Practice pg 252 #3-5, 8, 31, 33, 34

4 Quiz

4.6 – Using Congruent pg 259 # 3-8, 14-17

Triangles CPCTC

5 4.7 – Isosceles and pg # 267 # 7-17 20, 26, 46

Equilateral Triangles

Congruence

6 Using CPCTC

and overlapping triangles

7 Review

8 Test

Chapter 4: Congruent Triangles

Date _________ Block_________

#_________

1. Triangles can be classified by the number of congruent sides that they have….

2. Triangles can also be classified by their angles

3. Always, sometimes, never…

a. A right triangle is equiangular: _____________________

b. An acute triangle is equilateral: ____________________

c. An obtuse triangle is isosceles: _____________________

d. A right triangle is scalene: _________________________

4. The Triangle Sum Theorem (∆∑ Thm.)

a. The sum of the measures of the angles of a triangle is 180°

1 2 3 180m m m

1 2

3

Zero Congruent Sides Two+ Congruent Sides All 3 Congruent Sides

3 Acute Angles One Obtuse Angle One Right Angle All Angles

Triangle Sum Properties

Name:______________________________

__ Date _________ Block_________

#_________

Proof:

Given: ∆ ABC with angles 1, 2, and 3

Prove: 1 2 3 180m m m

1. ∆ ABC with angles 1, 2, and 3 1. Given

2. Line t is parallel to AB

2.

3. 4 1m m 3.

4. 4.

5. 4 3 5 180m m m 5.

6. 6. Substitution

7. 7.

5. What if the triangle is a right triangle, what must the two acute angles always add up to?

6. The ratio of the angles of a triangle is 3:6:9. Classify the triangle by its angles.

Statements Reasons

1 2

3

A B

C t

4 5

7. Exterior angles of a triangle – formed by __________________________ one of the sides.

What angle is supplementary to that angle?

8. Each exterior angle has two _________________________ interior angles

9. Find the degree measure of the exterior angles below…

10. Exterior Angles Theorem: The measure of an exterior angle is equal to

_________________________________________

70° 40°

80°

50°

1 2

3

Directions: Find the value of each variable.

Directions: Find the measure of each numbered angle

Closure: Compare and contrast the triangle sum theorem and triangle exterior angle thm.

Name:_______________________________

1.) ABC is an isosceles triangle such that each leg is 4cm less than twice the base. If the perimeter of the

triangle is 17cm, find the length of each side of the triangle.

2.) PQR is an equilateral triangle. One side measures 2x + 5 and another side measures x + 35. Find the

length of each side.

3.) The perimeter of ABC is 18m. If the second side is three times larger than the first side, and the third side

is three more than the first side, what is the measure of each side of the triangle?

4.) BCD is isosceles with C as the vertex angle. Find x and the measure of each side if BC = 2x + 4,

BD = x + 2, and CD = 10.

5.) HKT is equilateral. Find x and the measure of each side if HK = x + 7 and HT = 4x – 8.

6.) ABC is isosceles with A as the vertex angle. AC is five less than two times a number. AB is three more

than the number. BC is one less than the number. Find the measure of each side.

Use the distance formula to classify each triangle by the measures of its sides.

7.) ABC with vertices A(0, 6), B(3, 6), and C(3,0)

8.) PST with vertices P(4, 0), S(-2, 0), and T(1, 5)

Warmup: Determine if each pair of “objects” is congruent or not. Explain your choice!

1) 2)

3) 4)

5) 6)

7) 8)

Corresponding Parts of Congruence & Triangle Congruence

Name:______________________________

__ Date _________ Block_________

#_________

a) Points can be named in any

consecutive order

b) Each corresponding vertex must be in

the same order for each figure

Reminder: Congruent figures have the same __________________ & ___________________.

a. Each ____________________ (“matching”) side and angle of congruent figures will

also be ________________________!

Example:

Naming Congruent Figures

2. Example #2: Given the fact that ABCD EFGH , complete the following.

a. Rewrite the congruence statement in a different way.

b. Name all congruent angles

c. Name all congruent sides

Congruent Angles Congruent Sides

A

B

C

D E

V

W

X

Y Z

This chapter will deal with congruent triangles.

3. Formal Definition: Congruent Triangles

a. Two triangles are congruent if and only if their vertices can be matched up so that

the corresponding parts (angles & sides) of the triangles are congruent.

It means that if you put the two triangles on top of each other, they would match up perfectly

Triangle congruence works the same as it did for the pentagons, and for all polygons.

4. Given HIJ MNO . Name all congruent sides and all congruent angles.

Write the triangles congruent in two other ways.

5. If ABC XYZ , and 3 12m A x , 3 1m B x 44m X x , find x.

6. The Third Angle Theorem: If two angles of one triangle are congruent to two angles

of another triangle, the third angles are _____________ ______________________

7. What if we want to prove that polygons are congruent? What do we need to do?!

8. Because triangles only have three sides, we can take some shortcuts…

9. If all three sides are given, we call this __________________.

a. SSS Postulate: If 3 sides of one triangle are congruent to 3 sides of another

triangle, then the triangles are congruent.

10. If two sides and the angle BETWEEN those sides are given, we call this _______________.

a. SAS Postulate: 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.

* Included:

11. Using the postulates in proofs… Given the figure below, prove: YAX YBX

A B

X

Y

Key concept: Any time a side is shared, always think ______________________ property!!!

Example 1.

Given: ,IE GH EF HF and F is the midpoint of GI

Prove: EFI HFG

Given:𝑃𝑄 ∥ 𝑆𝑅 ; 𝑃𝑄 ≅ 𝑅𝑆

Prove: ∆PQS ≅ ∆RSQ

E

I

G

H

F

Warmup: Define the postulate below. Also, mark the triangles appropriately.

1. Angle Side Angle (ASA) Postulate:

2. Angle Angle Side (AAS) Theorem

a. If two angles and the non-included side of one triangle are congruent to two angles

and the corresponding non-included side of another triangle, then the triangles are

congruent.

Flow chart proof of AAS Theorem (remember: theorems are proven, postulates are accepted!)

Given: A X , B Y , and BC YZ Prove: ABC XYZ

Name:______________________________

__ Date _________ Block_________

#_________

ASA &AAS Triangle congruence

1. We’ve already briefly looked at the AAS theorem. Another method is the HL theorem.

a. H stands for _______________________.

b. L stands for ___________________.

2. Quick refresher: the hypotenuse is the side _______________________ the right angle.

Both of the other two sides are called legs.

3. The HL Theorem

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

parts of another right triangle, then the triangles are congruent.

b. Write a congruence statement: ____________________ * Review!

c. Identify six congruent parts in the space below.

4. For what values of x and y can the triangles be proven congruent by the HL theorem.

*** Only applies to right triangles!

D

E

F L

M

N

x x +3

3y

y +1

HL Right Triangle Congruence

Name:______________________________

__ Date _________ Block_________

#_________

5. Determine another side or angle that would have to be congruent to use the…

a. HL: ___________________________

b. SAS: __________________________

6. Given: 90m O m P , MN QR , OM PQ

Prove: MOR QPN

Statements Reasons

Closure: Answer the following.

1) Compare and contrast the HL theorem to the AAS theorem

2) When do you use the SAS postulate for a right triangle?

M

O

N

R

P

Q

A C

B

D

Ways to Prove Congruent Triangles

SSS SAS AAS ASA HL

Triangle Classifications that do NOT prove Congruent Triangles

AAA SSA

Understanding the term INCLUDED for Triangles

Given 2 Sides and 1 Angle Given 2 Angles and 1 Side

Name:______________________________

__ Date _________ Block_________

#_________

With a partner, describe the process for proving that two triangles are congruent. Be thorough in your

explanation. Imagine that you were explaining it to someone that was absent the last couple of days… Use

drawings, labels, or whatever you need in your explanation.

Name:_______________________________ Block:___________ Triangle Proofs

Complete each proof 1. Given: AB DC BC AD

Prove: ΔABC Δ CDA Proof:

Statement Reasons 1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

2. Given: 𝑙 𝑚 , PT ≅ QS , Q is the midpoint of PR Prove:Δ PQT Δ QRS Proof

Statement Reasons 1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

1 2

4 3 C

B

A D

l

m

R

Q

S

P T

1

2

3. Given: YZ bisects WX at P. WX bisects YZ at P. WY ≅ ZX

Prove: ΔWYP ΔXZP

Statement Reasons 1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

Given: AX XY , BY XY , Z is the midpoint of XY , 1 ≅ 2 Prove: ΔAXZ ΔBYZ

Statement Reasons 1. 2. 3. 4. 5. 6. 7. 8. 9.

1. 2. 3. 4. 5. 6. 7. 8. 9.

Y

W

P

Z

X

4 3

1 2

A

X Z Y

B

Given: AM = BM, Cm = DM Prove: ΔACM ΔBDM

Given: RS= RQ; ST = QT Prove: ΔRST ΔRQT

Statement Reasons 1. 2. 3. 4.

1. 2. 3. 4.

Given: M R, O is the midpoint of MR Prove: ΔMON ΔROP

Statement Reasons 1. 2. 3. 4. 5.

1. 2. 3. 4. 5.

Statement Reasons 1. 2. 3. 4.

1. 2. 3. 4.

A D

C B

M

Q

S

T R

P

M

R

O

N

Given: AC BC, m is the midpoint Prove: ΔACM ΔBCM

Statement Reasons 1. 2. 3. 4. 5.

1. 2. 3. 4. 5.

Given: B is the midpoint of AC ,

3 & 4 are right angles Prove: ΔABE Δ CBD

Statement Reasons 1. 2. 3. 4. 5. 6.

1. 2. 3. 4. 5. 6.

C

B A M

A

1 3

E

B

C

D

2 4

1. Complete the proof below.

Given: AC CE , BC CD

Prove: ACB ECD

2. Now that you know ACB ECD (hopefully you proved it!), list all of the congruent

sides and congruent angles. Think back to the first day of congruence if it will help you!

3. Look back at the triangles at the top of the page. What if the problem asked us to prove

AB DE instead of asking us to prove ACB ECD ? How could we do that?

4. Well, we could use the same strategy as we used for #2!

5. Once the triangles themselves are congruent, then all ___________________________ parts

of the triangles are also congruent!

a. We can use this as a ______________________ in our proofs. We call it…

A

B

C

D

E

Name:______________________________

__ Date _________ Block_________

#_________

Corresponding Parts of Congruence & Triangle Congruence

6. Example using CPCTC

Given: BCAB & BD is the angle bisector of ABC

Prove: AD CD

Game plan: First, prove that ABD CBD (we can use SSS, SAS, ASA, or AAS)

Next, conclude that AD BC using CPCTC!

Statements Reasons

1. BCAB

2. 2. Given

3. 3. Definition of an bisector

4. 4.

5. ABD CBD 5.

6. AD CD

7. Challenge: What does CPCTC stand for?

a. Say it 3 times fast…

8. Wrap Up: Explain what CPCTC means. When do you use it? Why is it helpful!?

A

B

C D

Definition: A triangles that has __________________________________ congruent sides.

1. Vocabulary Terms…

2. The Isosceles Triangle Theorem:

a. If two sides of a triangle are congruent, then the angles opposite those sides (the

base angles) are also congruent (proof on page 229).

3. The Converse of the Isosceles Triangle Theorem

a. If

Name:______________________________

__ Date _________ Block_________

#_________

Corresponding Parts of Congruence & Triangle Congruence

More Examples:

Example #1: Solve for x. The figure shown is a regular hexagon.

Example #2: Triangle ABC is isosceles with base AC.

2 8

4 20

m A x

m B x

What type of triangle is ABC, acute, obtuse, right, or equiangular?

Closure: Compare and contrast isosceles triangles with equilateral & equiangular triangles.

x

Sometimes it is easier to prove

triangles are congruent when

you separate them…

1. Given: AB DE & and are rt 'sBAD ADE Prove: ADB DAE

Examples:

STATEMENTS REASONS

Overlapping Triangles will sometimes

have “shared” sides or angles.

Make sure to state that they are

congruent using the _________________

property

A

B

D

E

A

B

D A D

E

Name:______________________________

__ Date _________ Block_________

#_________

Using CPCTC for Overlapping Triangles

2. Formal Proof: Redraw the triangles, add arc and tick marks and complete the proof.

Given: ABD CDB and CBD ADB Prove: AB CD

4.

A

B

C

D

E

1. a. Outline Δ SWU. b. outline Δ MIU with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. Which part appear to be corresponding? Complete the statement below sw and _________ S__________ SU and __________ SUW and _________ UW and_________ SWU and ________

2. a. Outline Δ MUW. b. outline Δ SUI with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent.

Which part appear to be corresponding? Complete the statement below sI and _________ S__________ SU and __________ SIU and _________ IU and_________ SUI and ________

1B) Outline the Δ’s again!! If ΔUWI is isosceles, what pair(s) of corresponding parts are congruent? Why?

2B) Outline the Δ’s again!! If ΔUWI is isosceles, what pair(s) of corresponding parts are congruent? Why?

U

S W I M

U

S W I M

U

S W I M

U

S W I M

3. a. Outline Δ BOA.

b. outline Δ STA with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent.

Which part appear to be corresponding? Complete the statement below BO and _________ O__________ BA and __________ OBA and _________ OA and_________ OAB and ________

If ΔBAS is isosceles, what pair(s) of corresponding parts are congruent? Why?

4. a. Outline Δ BSO. b. outline Δ SBT with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent. Which part appear to be corresponding? Complete the statement below BS and _________ O__________ BO and __________ OBS and _________ SO and_________ BSO and ________

If ΔBAS is isosceles, what pair(s) of corresponding parts are congruent? Why?

Outline the Δ’s again!! If ΔSUM is isosceles, what pair(s) of corresponding parts are congruent? Why?

Outline the Δ’s again!!

If ΔSUM is isosceles, what pair(s) of corresponding parts are congruent? Why?

U

S W I M

U

S W I M

T O

A

B S

T O

A

B S

5. a. Outline Δ HOS. b. outline Δ DNS with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent.

Which part appear to be corresponding? Complete the statement below HO and _________ H__________ HS and __________ HOS and _________ OS and_________ OSH and ________

6. a. Outline Δ HUN. b. outline Δ DUO with a different color. c. Mark any reflexive sides/ angles congruent. d. Mark any vertical angles congruent.

Which part appear to be corresponding? Complete the statement below HU and _________ U__________ UN and __________ H and _________ HN and_________ HNU and ________

Extra Practice

1. Given:𝐻𝐿 ≅ 𝐻𝑅 , 𝐻𝐴 ≅ 𝐻𝐷 Prove: L R

2. Given: LD ID, AI IN Prove: 1 2

O

U

N

D H

O

U

N

D H

H

A D

R L

L N

D A

I

3. Given: Fm FN, Dm ≅ HN, EF GF, DE HG Prove: ΔDEN ΔHGM

4. Given: WY XZ WZ ZY , XY ZY Prove ΔWYZ ΔXZY

Closure: Confidence Meter for the upcoming test? 1 2 3 4 5 6 7 8 9 10

Name the “good” and the “bad” …

E

N M D

G

H

F

X W

Y Z