Download Triangle Congruence

GO GIANTS!
 Pick up notes and the Exploration Activity
Tonight’s HW:
1. P 245 # 1-7
2. P 256 #1-8
3. Make notecards from U2L8 and U2L10 (definition one
side and vocab. word on the other side
Agenda
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Review Transformation Test
Exploration Activity!
4.4: SSS and SAS
4.5: AAS ASA HL
Proof Practice!
Stand Up!
Transformation Test Results
Overall, I am very proud of each and everyone of you for
putting your best effort into the test.
1st Period
o Average 27.1 out of 34
o 80%
6th Period
o Average 28.7 out of 31
o 84%
• You will get the tests back sometime this week. I still
have students who need to take it!
• The scores are on Infinite Campus
Learning Objective(s)
By the end of this period you will be able to:
① SWBAT prove triangles congruent by using SSS, SAS, ASA, AAS,
and HL
Exploration Activity
In order to prove that triangles are congruent you need to show
that:
1) All the angles are congruent
2) All the sides are congruent
However, there are five shortcuts! We will be investigating these
with your table-mates.
Exploration Activity
With your table, fill out the following worksheet.
You will need
1) A straightedge
2) A protractor
If you do not have the following, you will lose participation points.
Triangle Congruence Activity
Expectations:
• You will work as a table. Everyone must be on the same
problem.
• You as a team are responsible for keeping one another on
task.
o I do not want to hear off topic discussions.
• You will discuss with your tablemates.
o “What did you get for the angle measures?”
o “Why are my sides different from yours?”
SSS and SAS Congruence (4.4)
Instead of having to prove that all sides and angles are congruent in
order to prove that triangles are congruent, we are going to learn 5
shortcuts.
There are five ways to prove triangles are congruent:
1. SSS
2. SAS
3. ASA
4. AAS
5. HL
Right now, we are going to discuss SSS and SAS.
4-4 Triangle Congruence: SSS and SAS
Side–Side–Side Congruence (SSS)
• If the sides of one triangle are congruent to the sides of a
second triangle, then the triangles are congruent.
• We abbreviate Side-Side-Side Congruence as SSS.
What is a possible congruent statement for the figures?
• Examples
• Non-Examples
4-4 Triangle Congruence: SSS and SAS
Included Angle
• An angle formed by two adjacent
sides of a polygon.
• B is the included angle between
sides AB and BC.
Whiteboards
1. What is the included
angle between the
sides BC and CA?
2. What are the sides of
the included angle A?
Side-Angle-Side Congruence
Side–Angle–Side Congruence (SAS)
• 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.
What is the possible congruence statement for the figures?
Example/ Non-Examples
• Example
• Non-Example
4-4 Triangle Congruence: SSS and SAS
Example 1:
(a) Explain why ∆ABC  ∆DBC.
Use the following sentence frame:
It is given that ____  ____ and __  ______
By the ___________________________,
____ _____. Therefore ________  _________ by
________
4-4 Triangle Congruence: SSS and SAS
Example 1(b) :
Explain why ∆XYZ  ∆VWZ.
It is given that ____  ____ and __  ______
By the __________________________________________,
____ _____. Therefore ________  _________ by
________
Whiteboards
Explain why ∆ABC  ∆CDA.
It is given that ____  ____ and __  ______
By the ___________ ____________ of Congruence,
____ _____. Therefore ________  _________
by ________
Whiteboards
Explain why ∆ABC  ∆DBC.
I am not going to to give you the
sentence frame, but I still want you to use complete
sentences. Follow what you have on your notes.
It is given that BA  BD and ABC  DBC. By
the Reflexive Property of , BC  BC. So ∆ABC 
∆DBC by SAS.
Example 2: Verifying Triangle Congruence
Show that the triangles are congruent for the given value of the
variable.
∆MNO  ∆PQR, when x = 5.
PQ  MN, QR  NO, PR  MO
∆MNO  ∆PQR by SSS.
Whiteboards
Show that the triangles are congruent for the given value of the
variable.
∆STU  ∆VWX, when y = 4.
ST  VW, TU  WX, and T  W.
∆STU  ∆VWX by SAS.
4-4 Triangle Congruence: SSS and SAS
Example 3:
The Hatfield and McCoy families are feuding over some land.
Neither family will be satisfied unless the two triangular fields
are exactly the same size. You know that BC is parallel to AD
and the midpoint of each of the intersecting segments. Write
a two-column proof that will settle the dispute.
. Given: BC || AD, BC  AD
Prove: ∆ABC  ∆CDB
Proof:
Closure Questions
Which postulate, if any, can be used to prove the triangles
congruent? In one sentence tell why or why not the triangles
are congruent.
1.
2.
Math Joke of the Day
• What do you call a broken angle?
• A rectangle!
Change it to 4.5
• On top of your 4.4 Triangle Congruence: ASA<
AAS, and HL please change it to 4.5
4.5 Triangle Congruence: SSS and SAS
There are five ways to prove triangles are congruent:
1. SSS
earlier today ( or on Wednesday – per 6)
2. SAS
3. ASA
Today!
4. AAS
5. HL
Included Side
• Earlier, we learned what an included angle is.
What do you think an included side would be?
Included side
• common side of two consecutive angles in a
polygon.
4-4 Triangle Congruence: SSS and SAS
Angle–Side–Angle Congruence (ASA)
• 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
• . What is a possible congruent statement for the figures?
ASA
• Examples
• Non-Examples
Angle-Angle-Side Congruence
Angle-Angle-Side(AAS)
• If two angles and a non-included side of one triangle are
congruent to the corresponding angles and a side of a
second triangle, then the two triangles are congruent.
What is the possible congruence statement for the figures?
Example/ Non-Examples: AAS
• Example
• Non-Example
Example 1 (a)
Example 1:
(a) Explain why ∆UXV  ∆WXV.
It is given that ____  ____. ________ is a right angle so
______ is also a right angle by ______________.
Therefore, ____________  ____________. (add this sentence
frame into your notes)
By the __________________________________________,
____ _____. Therefore ________  _________ by
________
Example 1 (b)
Example 1:
(b) Explain why ∆ECS  ∆TRS.
This time, I am not going to give you a sentence frame, but I
still want you to use COMPLETE SENTENCES to explain why
the triangles are congruent.
Whiteboard
Determine if you can use ASA to prove
NKL  LMN. Explain.
By the Alternate Interior Angles Theorem. KLN  MNL. NL  LN by
the Reflexive Property. No other congruence relationships can be
determined, so ASA cannot be applied.
Whiteboard!
On your whiteboard, draw a right triangle
1. Label the legs of the triangle
2. Label the hypotenuse
Hypotenuse-Leg (HL) Congruence
Hypotenuse-Leg Congruence (HL)
• If the hypotenuse and a leg of a right triangle are congruent
to the hypotenuse and a leg of another right triangle, then
the triangles are congruent.
• IMPORTANT: The hypotenuse is ALWAYS across from the
right angle ( highlight this in your notes)
Examples/Non-Examples: HL
• Example
• Non-Example
Whiteboards
Identify the postulate or theorem that proves the triangles
congruent.
HL
ASA
SAS or SSS
Example 2
Proof Practice!
Whiteboard Flash!
I am going to show you two triangles
You are going to write down whether they are congruent by
SSS, SAS, AAS, ASA, or HL!
Once your entire table thinks they have it correct, STAND UP!
First table to have ALL their members stand up with the
correct statement wins that round.
Note: The triangles might not be congruent. If so, state they are not
congruent.
A) SSS
C) AAS
B) SAS
D) ASA
9)
A) ASA
C) SAS
B) AAS
D) SSS
A) Not congruent
C) SAS
B) ASA
D) SSS
10)
A) SSS
C) SAS
B) Not congruent
D) ASA
A) SAS
C) SSS
B) AAS
D) ASA
6)
A) Not congruent
C) ASA
8)
B) SSS
D) AAS
A) Not congruent
C) ASA
B) SSS
D) AAS
8)
A) Not congruent
C) SAS
10)
B) ASA
D) SSS
A) AAS
C) SSS
B) SAS
D) ASA
7)
A) SSS
C) AAS
9)
B) SAS
D) ASA
congruent
A) SSS
C) Not congruent
B) AAS
D) ASA
14)
B) SAS
D) ASA
A) SAS
C) AAS
16)
B) Not congruent
D) SSS
A) SAS
C) SSS
B) Not congruent
D) ASA
18)
A) AAS
C) SAS
B) Not congruent
D) SSS
A) ASA
C) Not congruent
B) AAS
D) SAS
19)
A) SAS
C) Not congruent
B) SSS
D) AAS
22)
B) AAS
D) SAS
A) Not congruent
C) AAS
24)
B) SAS
D) ASA
23)
A) AAS
C) Not congruent
B) SAS
D) SSS
A) Not congruent
C) AAS
B) SA
D) AS
24)
A) ASA
C) AAS
B) Not congrue
D) SAS
A) SSS
C) AAS
10)
B) ASA
D) SAS
A) AAS
C) Not congruent
B) ASA
D) SAS
SSS
ASA
uent
A) SSS
C) Not congruent
16)
B) ASA
D) SAS
A) Not congruent
C) ASA
B) SAS
D) AAS
A) LL
C) AAS
B) LA
D) ASA
12)
A) HL
C) SAS
B) HA
D) Not congruent
A) ASA
C) SAS
B) HA
D) AAS
13)
A) ASA
C) HA
15)
B) HL
D) SAS
A) SSS
C) HL
B) HA
D) ASA
23)
A) SAS
C) ASA
B) AAS
D) Not congruent
A) AAS
C) HL
B) LL
D) Not congr
7)
A) Not congruent
C) HA
B) H
D) S
11)
A) AAS
C) ASA
B) HA
D) LA
20)
A) SAS
C) ASA
22)
B) LA
D) LL
Part II: Missing Info
State what additional information is
required in order to know that the
triangles are congruent for the
reason given.