Download Lesson 22: Congruence Criteria for Triangles—SAS

Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
Lesson 22: Congruence Criteria for Triangles—SAS
Learning Target

I can explain why two triangles that satisfy the SAS congruence criterion must be congruent.
Discussion
Given angles ∠𝐴𝑂𝐵 and ∠𝐴′ 𝑂′ 𝐵 ′ , how can we tell whether they have the same degree without
having to measure each angle individually?
Quick Write:
What is congruence? ________________________________________________________________________
__________________________________________________________________________________________
__________________________________________________________________________________________
http://vimeo.com/45773877 ( 2.53 mins ) http://goo.gl/Vjg9IB ( 20-30 sec)
We are going to show that there are criteria that refer to a few parts (sides and/or angles) of two
triangles and a correspondence between them that guarantee triangle congruency (i.e., existence of
rigid motion/_____________).
Side-Angle-Side triangle congruence criteria (SAS):
Given two triangles △ 𝑨𝑩𝑪 and △ 𝑨′𝑩′𝑪′ so that
1. 𝑨𝑩 = 𝑨′𝑩′ (Side)
A
B
B'
2. 𝒎∠𝑨 = 𝒎∠𝑨′ (Angle)
3. 𝑨𝑪 = 𝑨′ 𝑪′ (Side).
C
A'
Then the triangles are congruent.
C'
Based on the previous definition for two triangles to be congruent, we have to identify a series of rigid
motion that map ∆𝐴′𝐵′𝐶′ to ∆𝐴𝐵𝐶.
GOAL:
To find a composition of rigid__________________ will map ∆𝐴′𝐵′𝐶′ to ∆𝐴𝐵𝐶.
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
1. The first thing we want to do is bring to the two triangles together at a common vertex.
A
B
B'
C
A'
C'
a. What should be the common vertex? ___________
b. What rigid motion should we use to bring those vertices together?
_______________________
c. Along which vector should the rigid motion be complete? __________
′ ′ ′
After the ________________(below), 𝑇𝐴′𝐴
⃗⃗⃗⃗⃗⃗⃗ (∆𝐴 𝐵 𝐶 ) shares one vertex with ∆𝐴𝐵𝐶, 𝐴. In fact, we can say
𝑇___________ (∆_____________) = ∆_____________.
2. Next we want to bring two corresponding sides together.
1. What two sides should we bring together? _____________________ or ____________________
2. What rigid motion should we use to bring those two sides together? ______________________
After a _____________________, 𝑅𝐴,−𝑚∠𝐶𝐴𝐶′′ (∆𝐴𝐵′′𝐶′′), a total of two vertices are shared with ∆𝐴𝐵𝐶, 𝐴
and 𝐶.
Therefore, 𝑅________(△ _____________) = △ _____________.
3. Finally, we want to map the triangles directly on top of each other. What rigid motion should we use to
map the triangles on top of each other? ____________________
Since a reflection is a rigid motion and it preserves angle measures, we know that 𝑚∠𝐵 ′′′ 𝐴𝐶 = 𝑚∠𝐵𝐴𝐶 and
⃗⃗⃗⃗⃗ .
so ⃗⃗⃗⃗⃗⃗⃗⃗⃗
𝐴𝐵′′′ maps to 𝐴𝐵
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
Write the transformations used to correctly notate the congruence (the composition of transformations)
that takes ∆𝐴′ 𝐵 ′ 𝐶 ′ ≅ ∆𝐴𝐵𝐶:
A
1st __________________________
B
B'
nd
2 __________________________
A'
C
3rd __________________________
Composite Notation ______(______(______(∆𝐴′ 𝐵 ′ 𝐶 ′ ) = ∆𝐴𝐵𝐶
C'
“Following Notation” ___________________
Congruence vs. Equality
When writing proofs and justifying steps, it’s important to use congruent and equal appropriately.
______________ are congruent.
______________ are equal.
Fill in the blanks with either = or 
AB _____ BC
AB ____ BC
mABC ______mDEF
ABC ______DEF
Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria:



Use a two-column proof to state which pairs of sides or angles are congruent/equal and why
If the triangles do meet the SAS congruence criteria, describe the rigid motion(s) that would map one
triangle onto the other.
When completing the proof you are looking to find a pair of sides congruent/equal, another pair of
sides congruent/equal, and then the pair of included angles congruent/equal.
**** In order to develop, you should remember the following:
1. Vertical angles have equal measure
2. Reflexive sides are congruent
3. Reflexive angles are congruent
4. Alternate interior angles and corresponding angles are congruent when two lines are parallel
5. Angle bisectors create congruent angles
6. Segment bisectors create congruent segments
5. Perpendicular lines create right angles
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Lesson 22
Period:________ Date:_________
M1
GEOMETRY
Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria and
describe the rigid motions.
Example 1. Given: 𝑚∠𝐿𝑁𝑀 = 𝑚∠𝐿𝑁𝑂, 𝑀𝑁 = 𝑂𝑁. Do △ 𝐿𝑁𝑀 and △ 𝐿𝑁𝑂 meet the SAS criteria? If
so, write a two column proof. If not, explain why.
a)
b) Describe the rigid motion: _______________________________________________________
̅̅̅̅ bisect each other. Do ∆𝐻𝐺𝐹 and ∆𝐾𝐺𝐿 meet the SAS criteria? If so, write
Example 2. Given: ̅̅̅̅
𝐹𝐿 and 𝐻𝐾
a two-column proof. If not, explain why.
a)
b) Describe the rigid motion: _______________________________________________________
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
GEOMETRY
Side-Angle-Side triangle congruence criteria (SAS)
Given two triangles △ 𝐴𝐵𝐶 and △ 𝐴′𝐵′𝐶′ such that:
1. 𝐴𝐵 = 𝐴′𝐵′ (Side)
2. 𝑚∠𝐴 = 𝑚∠𝐴′ (Angle)
3. 𝐴𝐶 = 𝐴′ 𝐶 ′ (Side).
Diagram:
Then the triangles are congruent.
****NOTICE: _________________________________________________________________________
Use your trasperancy to help you map the △ 𝐴′𝐵′𝐶′ to △ 𝐴𝐵𝐶.
A
B
B'
C
A'
C'
M1
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
Lesson 22: Congruence Criteria for Triangles—SAS
Classwork
Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria and
describe the rigid motion.
̅̅̅ bisect each other. Do ∆𝐾𝐿𝐽 and ∆𝑀𝐿𝑁 meet the SAS criteria? If so, write a two
1. Given: ̅̅̅̅̅̅
𝐾𝑀 and ̅𝐽𝑁
column proof. If not, explain why.
b) Describe the rigid transformation________________________________________________________
̅̅̅̅ ∥ 𝐶𝐷
̅̅̅̅ , 𝐴𝐵 = 𝐶𝐷. Do ∆𝐴𝐷𝐵 and ∆𝐶𝐵𝐷 meet the SAS criteria? If so, write a two column
2. Given: 𝐴𝐵
proof. If not, explain why.
b) Describe the rigid transformation: ______________________________________________________
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
̅̅̅̅, ̅̅̅̅̅
̅̅̅̅. Do ∆𝐽𝑀𝐿 and ∆𝐾𝐿𝑀 meet the SAS criteria? If so, write a
3. Given: 𝐽𝑀 = 𝐾𝐿, ̅̅̅̅̅
𝐽𝑀 ⊥ 𝑀𝐿
𝐾𝐿 ⊥ 𝑀𝐿
two column proof. If not, explain why.
STATEMENT
REASON
STATEMENT
REASON
4. Given: ̅̅̅̅
𝐵𝐹 ⊥ ̅̅̅̅
𝐴𝐶 , ̅̅̅̅
𝐶𝐸 ⊥ ̅̅̅̅
𝐴𝐵 .
Do △ 𝐵𝐸𝐷 and △ 𝐶𝐹𝐷 meet the SAS criteria?
Lesson 22
NYS COMMON CORE MATHEMATICS CURRICULUM
Name:___________________________________
Period:________ Date:_________
M1
GEOMETRY
Lesson 22: Congruence Criteria for Triangles—SAS
Homework
Directions: Develop a proof to justify whether the triangles meet the SAS congruence criteria:
When completing the following proofs you are looking to find a pair of sides congruent/equal, another
pair of sides congruent/equal, and then the pair of included angles congruent/equal.
****In order to find these pairs you should remember the following:
1. Vertical angles have equal measure
2. Reflexive sides are congruent
3. Reflexive angles are congruent
4. Alternate interior angles and corresponding angles are congruent when two lines are parallel
5. Angle bisectors create congruent angles
6. Segment bisectors create congruent segments
5. Perpendicular lines create right angles
1. Do ∆𝐺𝐻𝐼 and ∆𝐼𝐽𝐺 meet the SAS criteria? If so, write a two column proof. If not, explain why
2. Given: ̅̅̅̅
𝐴𝐸 bisects angle ∠𝐵𝐶𝐷, 𝐵𝐶 = 𝐷𝐶. Do ∆𝐴𝐵𝐶 and ∆𝐴𝐷𝐶 meet the SAS criteria? If so, write a
two column proof. If not, explain why.