Download Unit 5 – Triangle Congruence

Unit 5 – Triangle Congruence
Day
Classwork
Day
Homework
Thursday
11/3
Proving SAS through Rigid Motions
1
HW 5.1
Friday
11/4
Using SAS to Prove Triangles Congruent
CPCTC
2
HW 5.2
Monday
11/7
Proving ASA and SSS through Rigid Motions
3
HW 5.3
Tuesday
11/8
AAS, HL
AAA, SSA do not determine Congruence
Unit 5 Quiz 1
Adding and Subtracting Segments/Angles
4
HW 5.4
5
HW 5.5
Thursday
11/10
Overlapping Triangle proofs
Unit 5 Quiz 2
6
HW 5.6
Friday
11/11
No School
Monday
11/14
Double Triangle Proofs
7
HW 5.7
Tuesday
11/15
Review
Unit 5 Quiz 3
8
Review Packet
Wednesday
11/16
Review
9
Review Packet
Thursday
11/17
Unit 5 Test
10
Wednesday
11/8
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Day 1 – Proving SAS through Rigid Motions
Recall: We have agreed to use the word ‘congruent’ to mean ‘there exists a _________________________ of basic
rigid motions of the plane that maps one figure to the other.’
We are going to show that there are criteria that refer to a few parts of the two triangles and a
correspondence between them that guarantee congruency (i.e., existence of rigid motion). We start with
the Side-Angle-Side (SAS) criteria.
In the figure below, 2 pairs of sides are congruent and the included angles are congruent.
AB  ____________
AC  __________
There are 3 rigid motions that will map
symbols .
ABC to
A  ___________
A ' B ' C ' . Describe each rigid motion in words and
1.
2.
3.
A
B
B'''
C
2
Examples:
1.
Case
Diagram
Transformations Needed
A
Shared Side
B
B'''
C
B"
A
Shared Vertex
B
C
C"
2. Given: Triangles with a pair of corresponding sides of equal length and a pair of included angles of
equal measure. Sketch and label three phases of the sequence of rigid motions that prove the two
triangles to be congruent.
Transformation
Sketch
3
Directions: Justify whether the triangles meet the SAS congruence criteria; explicitly state which pairs of
sides or angles are congruent and why. If the triangles do meet the SAS congruence criteria, describe the
rigid motion(s) that would map one triangle onto the other.
1. Given: ∠𝐿𝑁𝑀 ≅ ∠𝐿𝑁𝑂, ̅̅̅̅̅
𝑀𝑁 ≅ ̅̅̅̅
𝑂𝑁.
Do △ 𝐿𝑀𝑁 and △ 𝐿𝑂𝑁 meet the SAS criteria?
̅.
2. Given: ∠𝐻𝐺𝐼 ≅ ∠𝐽𝐼𝐺, ̅̅̅̅̅
𝐻𝐺 ≅ 𝐽𝐼
Do △ 𝐻𝐺𝐼 and △ 𝐽𝐼𝐺 meet the SAS criteria?
3. Given: ̅̅̅̅
𝐴𝐵 ∥ ̅̅̅̅
𝐶𝐷, ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐶𝐷 .
Do △ 𝐴𝐵𝐷 and △ 𝐶𝐷𝐵 meet the SAS criteria?
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4. Given: 𝑚∠𝑅 = 25°, 𝑅𝑇 = 7", 𝑆𝑈 = 5", 𝑆𝑇 = 5".
Do △ 𝑅𝑆𝑈 and △ 𝑅𝑆𝑇 meet the SAS criteria?
̅̅ bisect each other.
5. Given: ̅̅̅̅̅
𝐾𝑀 and ̅̅
𝐽𝑁
Do △ 𝐽𝐾𝐿 and △ 𝑁𝑀𝐿 meet the SAS criteria?
̅̅̅̅ bisects angle ∠𝐵𝐶𝐷, ̅̅̅̅
6. Given: 𝐴𝐸
𝐵𝐶 ≅ ̅̅̅̅
𝐷𝐶 .
Do △ 𝐶𝐴𝐵 and △ 𝐶𝐴𝐷 meet the SAS criteria?
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Day 2 – Using SAS to Prove Triangles Congruent
Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SAS triangle
congruence criteria to prove the triangles are congruent.
1. Given:
̅̅̅̅ ≅ ̅̅̅̅
𝐴𝐵
𝐷𝐶
__________________________
__________________________
Prove:
2. Given:
△ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵.
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝑅𝑆
AB RS
__________________________
__________________________
Prove:
△ 𝐴𝐵𝐶 ≅△ 𝑅𝑆𝑇.
Example:
We already know that the base angles of an isosceles triangles are congruent. We are going to prove this
fact in two ways: (1) by using transformations, and (2) by using SAS triangle congruence criteria.
Prove Base Angles of an Isosceles are Congruent Using Transformations
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ .
Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵
Prove: ∠𝐵 ≅ ∠𝐶.
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Prove Base Angles of an Isosceles are Congruent Using SAS
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ .
Given: Isosceles △ 𝐴𝐵𝐶, with 𝐴𝐵
Prove: ∠𝐵 ≅ ∠𝐶.
Corresponding parts of congruent
triangles are congruent.
CPCTC
1. Given:
EA  AD
FD  AD
F
E
AC  DB
AE  DF
Prove:
EC  FB
A
B
C
D
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2. Given:
KM bisects
K
LKJ
LK  JK
Prove:
M is the midpoint of LJ
L
J
M
K
3. Given:
Prove:
KM  LJ
LM  JM
KLJ is isosceles
L
M
J
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4. Given:
C is a midpoint of AE
BC  DC
A
C
B
3 1
D
2 4
E
1 2
BF  DF
Prove:
AF  EF
F
T
5. Given:
Prove:
TR  TS
TM  TN
1 2
M
1
R
N
2
S
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Day 3 – Proving ASA and SSS through Rigid Motions
A second criteria that guarantees congruency (i.e., existence of rigid motion) is Angle-Side-Angle.
1. In the diagram below, describe the rigid motion that maps
congruent by SAS? Why or why not?
ABC to
ABC ''' . Will the triangles be
Since the triangles have two pairs of corresponding congruent angles and the included side
congruent, the triangles are congruent by Angle-Side-Angle.
2. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the ASA
triangle congruence criteria to prove the triangles are congruent.
Given:
ABC  DCB
__________________________
__________________________
Prove:
△ 𝐴𝐵𝐶 ≅△ 𝐷𝐶𝐵.
A
Given:
AD  BC
__________________________
__________________________
Prove:
△ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷
B
D
C
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3. In the diagram below, describe the rigid motion that maps
congruent? Why or why not?
ABC to AB ' C . Will the triangles be
Since the triangles have three pairs of corresponding congruent sides, the triangles are congruent
by Side-Side-Side.
4. Describe the additional piece(s) of information needed for each pair of triangles to satisfy the SSS
triangle congruence criteria to prove the triangles are congruent.
Given:
KM and JN bisect each other at L
__________________________
Prove:
△ 𝐽𝐾𝐿 ≅△ 𝑁𝑀𝐿.
Given:
R is the midpoint of KL
__________________________
__________________________
Prove:
△ 𝐽𝐾𝑅 ≅△ 𝐽𝐿𝑅
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1. Given:
Prove:
2. Given:
Prove:
̅̅̅̅, ∠𝐻 ≅ ∠𝑃.
𝑀 is the midpoint of 𝐻𝑃
GHM  RPM
Rectangle 𝐽𝐾𝐿𝑀 with diagonal ̅̅̅̅̅
𝐾𝑀.
JKM  LMK
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3. Given:
Prove:
Circles with centers 𝐴 and 𝐵 intersect at 𝐶 and 𝐷.
∠𝐶𝐴𝐵 ≅ ∠𝐷𝐴𝐵.
4. Given:
Prove:
∠𝑤
(1)
(2)
≅ ∠𝑥 and ∠𝑦 ≅ ∠𝑧.
△ 𝐴𝐵𝐸 ≅△ 𝐴𝐶𝐸.
̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐶 and ̅̅̅̅
𝐴𝐷 ⊥ ̅̅̅̅
𝐵𝐶 .
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Day 4 –AAS & HL Triangle Congruences
1. Consider a pair of triangles that meet the AAS criteria (see below). If you knew that two angles of one
triangle corresponded to and were equal in measure to two angles of the other triangle, what conclusions
can you draw about the third angles of each triangle?
Given this conclusion, which formerly learned triangle congruence criteria can we use to determine if the
pair of triangles are congruent?
Therefore, the AAS criterion is actually an extension of the
criterion.
triangle congruence
Practice: Using only the information given, decide whether the triangles are congruent by ASA or AAS.
1.
3.
AB  BC
DC  BC
2.
BC AD
AB DC
4.
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5. Consider the two right triangles below that have congruent hypotenuses and one pair of congruent legs.
Are the triangles congruent? Why?
This is our last criteria to prove triangles congruent: Hypotenuse-Leg (HL)
**Note that only RIGHT triangles can be proved congruent using HL**
Practice: Using only the information given, decide by which method the triangles could be proved
congruent.
ED  BA
1.
2.
3. HG  HI
4. E is the midpoint of CD .
A and B are right angles
AC  DF
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Criteria that do not determine two triangles as congruent:
Two sides and a non-included angle (SSA): Observe the diagrams below. Each triangle has a set of
adjacent sides of measures 11 and 9, as well as the non-included angle of 23˚. Yet, the triangles are not
congruent.
Examine the composite made of both triangles. The sides of lengths 9 each have been dashed to show
their possible locations.
The pattern of SSA cannot guarantee congruence criteria. In other words, two triangles under SSA criteria
might be congruent, but they might not be; therefore we cannot categorize SSA as congruence criterion.
Practice: Decide if the triangles are congruent. If so, by what method?
1. Given:
A  X , BA  YX , ZX  CA
2. Given: CA  ZX , BC  YZ ,
A X
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Three Congruent Angles (AAA): Observe the diagrams below. What is the measure of the missing
angles? Notice that the triangles have 3 pairs of congruent angles. Yet, the triangles are not congruent.
45°
45°
The pattern of AAA cannot guarantee congruence criteria. In other words, two triangles under AAA
criteria might be congruent, but they might not be; therefore we cannot categorize AAA as congruence
criterion.
B
1.
Given: BD  AC
A C
Prove:
ABD  CBD
A
X
1
P
2.
Given: PQ
RS
QO  RO
Prove: XO  YO
C
D
Q
O
R
2
Y
S
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ADDING & SUBTRACTING SEGMENTS & ANGLES
If a pair of congruent segments/angles are added to another pair of congruent segments/angles, then the
resulting segments/angles are congruent. Similarly, if a pair of congruent segments/angles are subtracted
from a pair of congruent segments/angles, then the resulting segments/angles are congruent.
Statements
Reasons___________
1. AB  CD
1. Given
2. BC  BC
2. Reflexive Property
3. AB  BC  CD  BC
3. Addition Property
4. AC  AB  BC
BD  CD  BC
4. A whole =’s the sum of its
parts
5. AC  BD
5. Substitution
Statements
Reasons___________
1. AC  BD
1. Given
2. BC  BC
2. Reflexive Property
3. AC  AB  BC
BD  CD  BC
3. A whole =’s the sum of its
parts
4. AB  BC  CD  BC
4. Substitution
5. AC  BD
5. Subtraction Property
Examples
1.
18
2.
3.
4.
19
5.
6.
7.
20
8. Given: PS  UR
PQ  UT
QR  TS
Prove:
Q T
9. Given:
1 4
2 3
Prove:
5
6
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Day 6 & 7 – Overlapping and Double Triangle Proofs
1. Given:
Prove:
̅̅̅̅ ⊥ 𝐵𝐶
̅̅̅̅ , 𝐵𝐶
̅̅̅̅ ⊥ 𝐷𝐶
̅̅̅̅
𝐴𝐵
̅̅̅̅
𝐷𝐵 bisects ∠𝐴𝐵𝐶, ̅̅̅̅
𝐴𝐶 bisects ∠𝐷𝐶𝐵
̅̅̅̅
̅̅̅̅ ≅ 𝐸𝐶
𝐸𝐵
△ 𝐵𝐸𝐴 ≅ ∆𝐶𝐸𝐷
2. Given: ̅̅̅̅
𝐵𝐹 ⊥ ̅̅̅̅
𝐴𝐶 , ̅̅̅̅
𝐶𝐸 ⊥ ̅̅̅̅
𝐴𝐵
̅̅̅̅ ≅ ̅̅̅̅
𝐴𝐸
𝐴𝐹
Prove: △ 𝐴𝐶𝐸 ≅ ∆𝐴𝐵𝐹
22
̅̅̅ ≅ 𝑌𝐾
̅̅̅̅ , 𝑃𝑋
̅̅̅̅ ≅ 𝑃𝑌
̅̅̅̅ , ∠𝑍𝑋𝐽 ≅ ∠𝑍𝑌𝐾
3. Given: 𝑋𝐽
Prove: ̅̅̅
𝐽𝑌 ≅ ̅̅̅̅
𝐾𝑋
̅̅̅ ≅ 𝐽𝐿
̅ , 𝐽𝐾
̅̅̅ ∥ 𝑋𝑌
̅̅̅̅
4. Given: 𝐽𝐾
̅̅̅̅ ≅ ̅̅̅̅
Prove: 𝑋𝑌
𝑋𝐿
23
5. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4
Prove: ̅̅̅̅
𝐴𝐶 ≅ ̅̅̅̅
𝐵𝐷
6. Given: ∠1 ≅ ∠2, ∠3 ≅ ∠4, ̅̅̅̅
𝐴𝐵 ≅ ̅̅̅̅
𝐴𝐶
Prove: (a) △ 𝐴𝐵𝐷 ≅△ 𝐴𝐶𝐷
(b) ∠5 ≅ ∠6
24
̅̅̅̅ ≅ 𝐴𝐶
̅̅̅̅ ,
7. Given: 𝐴𝐵
̅̅̅̅
𝑅𝐵 ≅ ̅̅̅̅
𝑅𝐶 ,
̅̅̅̅ ≅ 𝑆𝐶
̅̅̅̅
Prove: 𝑆𝐵
̅ , ̅̅̅
8. Given: ̅̅̅
𝐽𝐾 ≅ 𝐽𝐿
𝐽𝑋 ≅ ̅̅̅
𝐽𝑌
̅̅̅̅ ≅ 𝐿𝑌
̅̅̅̅
Prove: 𝐾𝑋
25
̅̅̅̅ ⊥ ̅̅̅̅
̅̅̅̅ ⊥ 𝐵𝑅
̅̅̅̅,
9. Given: 𝐴𝐷
𝐷𝑅, 𝐴𝐵
̅̅̅̅
𝐴𝐷 ≅ ̅̅̅̅
𝐴𝐵
Prove: ∠𝐷𝐶𝑅 ≅ ∠𝐵𝐶𝑅
26