Download GETE0403

Triangle Congruence
by ASA and AAS
4-3
4-3
1. Plan
GO for Help
What You’ll Learn
Check Skills You’ll Need
• To prove two triangles
In k JHK, which side is included between the given pair of angles?
congruent using the ASA
Postulate and the AAS
Theorem
1. &J and &H JH
In k NLM, which angle is included between the given pair of sides?
3. LN and LM lL
4. NM and LN lN
To prove that the two sides of
a lacrosse goal are congruent
triangles, as in Example 2
Give a reason to justify each statement.
5. PR > PR
P
6. &A > &D
Q
R
Reflexive Prop. of O
A
1
2
3
4
D
Angle-Side-Angle (ASA) Postulate
P
B
B
A
1
4
5
A
A
B
B
D
C
Multiple Choice Which triangle is congruent to #CAT by the ASA Postulate?
E
D
C
See p. 196E for a list of the
resources that support this lesson.
Using ASA
E
D
C
B
EXAMPLE
Lesson Planning and
Resources
H
E
D
C
B
A
3
C
B
A
2
1
E
D
More Math Background: p. 196C
N
K
G
#HGB > #NKP
C
Using ASA
Real-World Connection
Planning a Proof
Writing a Proof
ASA is presented in this lesson
as a postulate, but it could be
established as a theorem (whose
proof requires constructing
congruent segments) that follows
from the SAS postulate, much
as SSS also could be established
as a theorem that follows from
the SAS Postulate. The proof
of the AAS Theorem follows
from the ASA Postulate and the
Triangle Angle-Sum Theorem.
In Lesson 4-2 you learned that two triangles are congruent if
two pairs of sides are congruent and the included angles are congruent (SAS).
The Activity Lab on page 204 suggests that two triangles are also congruent if
two pairs of angles are congruent and the included sides are congruent (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 two triangles are congruent.
To prove two triangles
congruent using the ASA
Postulate and the AAS
Theorem
Math Background
Using the ASA Postulate and the AAS Theorem
Postulate 4-3
1
Examples
If 2 ' of a
k are O to
C
B F
E
2 ' of another k, the third ' are O.
S
Key Concepts
Objectives
2. &H and &K HK
. . . And Why
1
Lesson 4-2
D
E
E
Test-Taking Tip
#DOG
#GDO
In a triangle
congruence statement,
remember to list
corresponding vertices
in the same order.
PowerPoint
#INF
#FNI
D
O
&C > &G, CA > GD, and &A > &D.
#CAT > #GDO by ASA. Choice C
is correct.
Quick Check
Bell Ringer Practice
C
F
G
A
T
Check Skills You’ll Need
For intervention, direct students to:
N
I
1 Can you conclude that #INF is congruent to either of the other two triangles?
Explain. No; the O side is not the included side.
Using the SAS and SSS Postulates
Lesson 4-2: Example 2
Extra Skills, Word Problems, Proof
Practice, Ch. 4
Proving Triangles Congruent
Chapter 4 Triangle Congruence by ASA and AAS
Special Needs
L1
Have students mark
congruent sides and
angles as shown to
distinguish between
AAS and ASA.
A
A
A
A
learning style: visual
learning style: verbal
S
A S
A S
Lesson 4-1: Example 4
Extra Skills, Word Problems, Proof
Practice, Ch. 4
L2
Students may substitute specific measures for the
congruent angles in the flow proof of the AAS
Theorem to see the dependence on the ASA Postulate
before proceeding to the general proof.
S
A
Below Level
A
213
213
2. Teach
Real-World
Guided Instruction
2
EXAMPLE
The Iroquois Nationals compete
for the World Lacrosse
Championship every four years.
2
EXAMPLE
Real-World
Connection
C
Lacrosse Study what you are given and
what you are to prove about the lacrosse goal.
Then write a proof that uses ASA.
Diversity
D
A
B
E
Front view of lacrosse goal
Given: &CAB > &DAE, AB > AE,
&ABC and &AED are right angles.
Lacrosse originated as
“baggataway,” a Native American
game with up to 200 players on
each side. Its modern name comes
from French Canadians.
Prove: #ABC > #AED
Proof: &ABC > &AED because all right angles are congruent.
You are given that AB > AE and &CAB > &DAE.
Thus, #ABC > #AED by ASA.
Teaching Tip
Examine the flow proof of the
AAS Theorem as a class. Point out
the three arrows leading to the
conclusion. Ask: Why are there
three arrows? Each arrow is
from one of the congruence
statements needed to prove the
AAS Theorem.
3
Here is how you can use the ASA Postulate in a proof.
Connection
Quick Check
2 Write a proof.
L
P
Given: NM > NP, &M > &P
N
M
O
It is given that NM O NP and lM O lP. lMNL O lPNO
because vert. ' are O. kNML O kNPO by ASA.
You can use the ASA Postulate to prove the Angle-Angle-Side Congruence
Theorem. A flow proof is shown below.
Prove: #NML > #NPO
EXAMPLE
Key Concepts
Point out that an effective Plan
for Proof may involve the strategy
working backward. In this Plan
for Proof, the conclusion of the
conditional statement is what
must be proved. The second
sentence of the Plan for Proof
tells why the hypothesis is true.
Theorem 4-2
Angle-Angle-Side (AAS) Theorem
If two angles and a nonincluded side of one
triangle are congruent to two angles and the
corresponding nonincluded side of another
triangle, then the triangles are congruent.
C
M
#CDM > #XGT
PowerPoint
Proof
Additional Examples
G
Prove: #ABC > #XYZ
Suppose that &F is congruent
to &C and &I is not congruent to
&C. Name the triangles that are
congruent by the ASA Postulate.
kFNI O kCAT O kGDO
A X
Given
B Y
Given
2 Write a paragraph proof.
A
BC YZ
Given
Y
X
Proof of the Angle-Angle-Side Theorem
Given: &A > &X, &B > &Y,
BC > YZ
1 Use the triangles in Example 1.
T
D
P
X
A
C Z
If 2 s of one are to 2 s of another ,
then the 3rd s are .
Y
B
Z
C
ABC XYZ
ASA Postulate
X
A proof starts in your head with a plan. It may help to jot down your plan. You will
find this especially helpful as proofs become more complex. As you become skilled
at proof, you will find that a good plan looks a lot like a paragraph proof.
B
Given: &A &B, AP BP
Prove: APX BPY
lAPX O lBPY by the
Vertical Angles Theorem, so
kAPX O kBPY by ASA.
214
Chapter 4 Congruent Triangles
Advanced Learners
English Language Learners ELL
L4
After students read the ASA Postulate and the AAS
Theorem, ask: If you could use only one in the
remainder of this course, which would you choose
and why?
214
learning style: verbal
Point out that AAS, or Angle-Angle-Side Theorem,
is a theorem and not a postulate. A postulate is an
accepted statement of fact, but theorems are
conjectures that are proven.
learning style: verbal
3
EXAMPLE
3 Write a Plan for Proof that
uses AAS.
Planning a Proof
A
Study what you are given and what you are to prove.
Then plan a proof that uses AAS.
Given: &S > &Q,
RP bisects &SRQ.
P
D
Prove: #SRP > #QRP
S
Prove: ABC CDA Because
AB n CD, lBAC O lDCA (alt. int.
angles). kABC O kCDA by
AAS if AB O CD, BC O DA,
or AC O AC. By Reflexive Prop.,
AC O AC.
R
The second statement is true by the Reflexive Property of Congruence.
Proof
3 Use the plan from Example 3 and write a proof.
See back of book.
4
EXAMPLE
4 Write a two-column proof of
Additional Example 3.
1. lB O lD, AB n CD (Given)
2. lBAC O lDCA (If lines are n,
then alt. int. angles are O.)
3. AC O AC (Reflexive Prop.
of O)
4. kABC O kCDA (AAS)
Writing a Proof
Study what you are given and what you are to prove.
Then write a proof that uses AAS.
nline
Q
Given: XQ 6 TR, XR bisects QT.
Prove: #XMQ > #RMT
Plan:
Visit: PhSchool.com
Web Code: aue-0775
X
XQ 6 TR gives two pairs of congruent
alternate interior angles. XR bisects QT
means QM > TM. Use AAS.
Statements
Quick Check
M
R
Resources
• Daily Notetaking Guide 4-3 L3
• Daily Notetaking Guide 4-3—
L1
Adapted Instruction
T
Reasons
1. XQ 6 TR
1. Given
2. &Q > &T, &X > &R
2. 9
3. XR bisects QT.
3. Given
4. QM > TM
4. Definition of segment bisector
5. #XMQ > #RMT
5. AAS
Closure
4 a. Supply the reason that justifies Step 2. If n lines, then alt. int. ' are O.
b. Critical Thinking Explain how you could prove #XMQ > #RMT by ASA.
kXMQ O kRMT because vert. ' are O.
EXERCISES
C
Given: &B &D, AB 6 CD
Q
Plan: #SRP > #QRP by AAS
if SP > QP or RP > RP.
Quick Check
B
For more exercises, see Extra Skill, Word Problem, and Proof Practice.
Explain why the letters of ASA
and AAS are written in a different
order. ASA compares triangles in
which O sides are between the
two pairs of O angles, and AAS
compares triangles in which O
sides are not between pairs of O
angles.
Practice and Problem Solving
A
Practice by Example
Example 1
GO for
Help
Name two triangles that are congruent by the ASA Postulate.
1. P
Q
(page 213)
S
X
W
2.
E
F
B
H
D
U
V
C G
A
T
kACB O kEFD
kPQR O kVXW
Answer each question without drawing the triangle.
R
I
3. Which side is included between &R and &S in #RST? RS
4. Which angles include NO in #NOM? lN and lO
Lesson 4-3 Triangle Congruence by ASA and AAS
215
215
5. Developing Proof Complete the proof by filling in the blanks.
Example 2
3. Practice
(page 214)
Given: &LKM > &JKM,
&LMK > &JMK
Assignment Guide
K
Prove: #LKM > #JKM
L
J
M
Proof: &LKM > &JKM and &LMK > &JMK are
1 A B 1-26
C Challenge
27-32
Test Prep
Mixed Review
33-36
37-43
given. KM > KM by the a. 9 Property of Congruence. Reflexive
#LKM > # JKM by the b. 9 Postulate. ASA
Proof
Homework Quick Check
6. Given: &BAC > & DAC
AC ' BD
7. Given: QR > TS, QR 6 S T
Prove: #QRT > #TSQ
Prove: #ABC > #ADC
6–7. See back of book.
B
To check students’ understanding
of key skills and concepts, go over
Exercises 5, 11, 17, 22, 26.
A
R
Q
C
T
S
D
Teaching Tip
Students practice several different
methods of writing and planning
proofs in this lesson. Point out
that a good plan uses only the
necessary information and clearly
states the reasons. The goal is
to plan and write clear and
correct proofs. Learning multiple
techniques will help students
reach that goal.
8. Developing Proof Complete the proof plan by filling in the blanks.
Example 3
(page 215)
Given: &UWT and &UWV are right angles,
&T > &V.
U
Prove: #UWT > #UWV
T
Plan: #UWT > #UWV by AAS if &T > &V,
&UWT > a. 9, and UW > b. 9. lUWV; UW
V
W
&UWT > &UWV because all c. 9 angles are congruent. right
UW > UW by the d. 9 Property of Congruence. Reflexive
Proof
9. Use your plan from Exercise 8 and write a proof. See back of book.
10. Developing Proof Complete the proof by filling in the blanks.
Example 4
(page 215)
Given: &N > &S, line / bisects TR at Q.
ᐉ
N
Prove: #NQT > #SQR
R
Q
T
GPS Guided Problem Solving
Statements
L3
L4
Enrichment
1.
2.
3.
c.
5.
L2
Reteaching
L1
Adapted Practice
Practice
Name
Class
L3
Date
Practice 4-3
Triangle Congruence by ASA and AAS
Tell whether the ASA Postulate or the AAS Theorem can be applied
directly to prove the triangles congruent. If the triangles cannot be
proved congruent, write not possible.
L
1.
R
Q
T
2.
U
Proof
3. W
X
M
N
S
B
4.
D
Z
Y
P
V
A
H
G
5.
N
6.
P
R
J
E
F
M
K
U
7.
V
W
Q
L
8. Z
9.
A
H
&N > &S
&NQT > &SQR
O bisects TR at Q.
9 TQ O QR
#NQT > #SQR
1.
2.
3.
4.
5.
Given
9 Vert. ' are O.
9 Given
Definition of bisect
9 AAS
11. Given: &V > &Y,
WZ bisects &VWY.
Prove: #VWZ > #YWZ
11–12. See back of book.
W
S
C
D
Y
C
X
B
G
R
Q
F
© Pearson Education, Inc. All rights reserved.
11. Write a flow proof.
10. Write a two-column proof.
Given: K M, KL ML
Given: Q S, TRS RTQ
Prove: JKL PML
Prove: QRT STR
K
J
Q
R
T
S
V
L
M
P
What else must you know to prove the triangles congruent for the
reason shown?
12. ASA
13. AAS
A
B
D
C
E
14. ASA
K
J
L
M
G
216
Q
N
F
H
12. Given: PQ ' QS, RS ' QS,
T is the midpoint of PR.
Prove: # PQT > #RST
E
T
S
Reasons
P
216
Chapter 4 Congruent Triangles
Z
T
Y
P
S
Alternative Method
B
Apply Your Skills
Write a congruence statement for each pair of triangles. Name the postulate or
theorem that justifies your statement.
kZVY O kWVY; AAS
V
W
M
T
13.
14.
15.
U
P
R
N
O
kPMO O kNMO; ASA
Real-World
16. Multiple Choice For the triangles shown,
&D > &T, &E > &U, and EO > UX.
Which of the following statements is true? D
#TUX > #DOE by ASA
#UTX > #DEO by AAS
#TXU > #ODE by ASA
#TUX > #DEO by AAS
Connection
23. kBDH O kFDH by
ASA since lBDH O
lFDH by def. of l bis.
and DH O DH by the
Reflexive Prop. of O.
26. Answers may vary.
Sample:
C
A
D
Proof
nline
Homework Help
Visit: PHSchool.com
Web Code: aue-0403
D
E
X
Exercise 11 Students sometimes
Prove: #MON > #QOP
Prove: #FGJ > #HJG
M
F
N
triangle must be flipped over in
order for the triangles to match
exactly.
G
Proof
J
P
22. Given: AE 6 BD, AE > BD,
&E > &D
GPS
Prove: #AEB > #BDC
E
Exercise 22 Because the parallel
segments terminate at the
transversal AC , students may have
difficulty spotting corresponding
angles. Have them copy just the
parallel segments AE and BD and
the transversal AC to help them
find the corresponding angles
more easily.
Prove: #BDH > #FDH
D
1 2
H
C
B
Visual Learners
H
23. Given: DH bisects &BDF,
&1 > &2.
D
B
F
Connection to Discrete Math
24. Constructions Using a straightedge, draw a triangle. Label it # JKL. Construct
#MNP > # JKL so you know that the triangles are congruent by ASA.
See back of book.
A
B
Proof 25. Given: AB 6 DC, AD 6 BC
D
think an angle bisector also
bisects the opposite side or is
a perpendicular bisector. Ask:
From the Given, can you conclude
&PQS &RQS or PQ RQ? no
Review how angle bisectors,
segment bisectors, and
perpendicular bisectors
are different.
Exercise 16 Point out that one
O
Q
to find a second way to complete
the proof. Ask: Suppose that you
did not know the Vertical Angles
Theorem. How could you find
another pair of congruent angles?
Because lN O lS, NT n RS by
the Converse of the Alt. Int. Angles
Thm., so lT O lR because they
are alt. int. angles. QT O QR by
definition of a bisector, and
kNQT O kSQR by AAS.
Error Prevention!
O
21. Given: &F > &H, FG 6 JH
Prove: #ABC > #CDA
See back of book.
GO
T
U
20. Given: &N > &P, MO > QO
A
B
Y
19. Reasoning Can you prove the triangles at the right
congruent using ASA or AAS? Justify your answer.
No; also need one pair of corres. sides O.
20. kMON O kQOP by
AAS; lMON and
lQOP are O vert. '.
22. kAEB O kBDC by
ASA, since lEAB O
lDBC because n lines
have O corr. '.
Z
17. Writing Anita says that you can rewrite any proof that uses the AAS Theorem
as a proof that uses the ASA Postulate. Do you agree with Anita? Explain.
See margin.
18. &E > &I and FE > GI. What else must you know to prove #FDE > #GHI
by AAS? By ASA? lFDE O lGHI; lDFE O lHGI
Congruent lapel and collar
triangles help you look sharp.
21. kFGJ O kHJG by
AAS since lFGJ O
lHJG because when
lines are n, then alt.
int. ' are O and GJ O
GJ by the Reflexive
Prop. of O.
S
kUTS O kRST; AAS
Exercise 10 Challenge students
Exercise 31 There are 20 ways
because 6C3 = 63 ?? 52 ?? 41 = 20. To
list the sets of three efficiently,
suggest that students rename
the congruence statements
1, 2, 3, 4, 5, and 6.
C
26. Reasoning If possible, draw two noncongruent triangles that have two pairs
of congruent angles and one pair of congruent sides. If this is not possible,
explain why. See left.
Lesson 4-3 Triangle Congruence by ASA and AAS
217
17. Yes; if 2 ' of a k are O
to 2 ' of another k,
then the 3rd ' are O.
So, an AAS proof can be
rewritten as an ASA
proof.
217
4. Assess & Reteach
C
Challenge
PowerPoint
Lesson Quiz
1. Which side is included between
&R and &F in FTR? RF
2. Which angles in STU include
US? lS and lU
Tell whether you can prove the
triangles congruent by ASA or
AAS. If you can, state a triangle
congruence and the postulate or
theorem you used. If not, write
not possible.
3.
G
Real-World
Connection
The two triangles above are
congruent if just one
additional condition is met.
H
I
P
Q
R
P
L
U
32.
9
5
30°
Y
A
What additional information
about KQ and NR will allow
you to conclude that
J
# JKQ > #MNR? Explain.
Q
They are l bisectors; ASA.
31. Probability Here are six congruence
statements about the triangles at the right. 13
20
&A > &X
&B > &Y
&C > &Z
AB > XY
AC > XZ
BC > YZ
L
M
W
V
P
R
A
32. #RST at the right has RS = 5, RT = 9,
and m&T = 30. Show that there is no SSA
congruence rule by constructing #UVW with
UV = 5, UW = 9, and m&W = 30, but with
#UVW R #RST. See left.
C
B
X
There are 20 ways to choose a group of three
statements from these six. What is the probability
that three statements chosen at random from the
six will guarantee that the triangles are congruent?
kGHI O kPQR; AAS
4.
27. a. Open-Ended Draw a triangle. Draw a second triangle that shares a common
side with the first one and is congruent to it. Check students’ work.
b. Think about how you drew your second triangle. What postulate or theorem
did you use to make the second triangle congruent to the first one?
most likely ASA
Use the figure at the right. Name as many pairs of
B
C
congruent triangles as you can for the information given.
E
28. ABCD is a parallelogram. 28–29.
See margin.
29. ABCD is a rectangle.
A
D
K
N
30. Reasoning #JKL > #MNP.
Z
Y
R
9
5
30
T
S
not possible
A
5.
Test Prep
Multiple Choice
B
X
C
34. Suppose RT > ND and &R > &N. What additional information is needed to
prove #RTJ > #NDF by ASA? F
F. &T > &D
G. &R > &N
H. & J > &D
J. &T > &F
kABX O kACX; AAS
Short Response
Alternative Assessment
Have students explain the four
ways they have learned to prove
triangles congruent—SSS, SAS,
ASA, and AAS—and include a
diagram with each method.
Extended Response
218
35. PQ bisects &RPS and &RQS. Justify each answer.
a. Which pairs of angles, if any, are congruent?
b. By what theorem or postulate can you prove
that #PRQ > #PSQ? See margin.
36. LJ 6 KG and M is the midpoint of LG.
a. Why is LM > GM? a–c. See margin, p. 219.
b. Can the two triangles be proved congruent
by ASA? Explain.
c. Can the two triangles be proved congruent
by AAS? Explain.
R
P
Q
S
L
K
M
J
G
Chapter 4 Congruent Triangles
28. kAEB O kCED,
kBEC O kDEA,
kABC O kCDA,
kBAD O kDCB
218
33. Which of the following is NOT a method used to prove triangles congruent? D
A. AAS
B. ASA
C. SAS
D. SSA
29. kAEB O kCED,
kBEC O kDEA,
kABC O kCDA,
kABD O kDCA,
kBAD O kDCB,
kABD O kDCB,
kCBA O kDAB,
kBCD O kADC
35. [2] a. lRPQ O lSPQ,
lRQP O lSQP
(Def. of l bisector)
b. ASA
[1] one part correct
Test Prep
Mixed Review
GO for
Help
Lesson 4-2
In Exercises 37 and 38, decide whether you can use the SSS Postulate or the SAS
Postulate to prove the triangles congruent. If so, write the congruence statement
and name the postulate. If not, write not possible.
L
37.
P
38.
M
T
O
Lesson 3-2
Lesson 1-9
R
Q
Resources
For additional practice with a
variety of test item formats:
• Standardized Test Prep, p. 253
• Test-Taking Strategies, p. 248
• Test-Taking Strategies with
Transparencies
S
not possible
N
kONL O lMLN; SAS
39. For any #ABC, which sides are not included between &A and &B?AC and
CB
40. State the theorem or postulate
a
that justifies the statement:
1
2
If &1 > &3, then a 6 b.
b
3
If corr. ' are O, then the lines are n.
Photography You want to arrange class-trip photos without overlap to make a
2 ft-by-3 ft poster. You collect 3 in.-by-5 in. and 4 in.-by-6 in. photos. What is the
greatest number of each type of photo that you can fit on your poster?
41. 3 in.-by-5 in. 56
Use this Checkpoint Quiz to check
students’ understanding of the
skills and concepts of Lessons 4-1
through 4-3.
Resources
Grab & Go
• Checkpoint Quiz 1
42. 4 in.-by-6 in. 36
43. What percent more paper is used for a large photo than a regular photo?
60% more paper
Checkpoint Quiz 1
Lessons 4-1 through 4-3
1. #RST > # JKL. List the three pairs of congruent corresponding sides and the
three pairs of congruent corresponding angles.
RS O JK ; ST O KL ; RT O JL ; lR O lJ; lS O lK; lT O lL
State the postulate or theorem you can use to prove the triangles congruent. If the
triangles cannot be proven congruent, write not possible.
2.
ASA
3.
4. SAS
SSS
36. [4] a. Def. of midpt.
5.
6.
7.
AAS
not possible
not possible
Use the information given in the diagram.
Tell why each statement is true.
8. &H > &K
L
K
c. Yes; if two ' of
one k are O to 2
' of another k,
the third ' are O.
N
If n lines, then alt. int. ' O.
9. &HNL > &KNJ Vert. ' are O.
10. #HNL > #KNJ ASA or AAS
lesson quiz, PHSchool.com, Web Code: aua-0403
H
J
Lesson 4-3 Triangle Congruence by ASA and AAS
b. Yes; lJLM O
lKGM because
they are alt. int. '
of n lines, and
lLMJ O lGMK
because vertical
' are O. So the >
are O by ASA.
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[3] incorrect ' for part
b or c, but otherwise
correct
[2] correct conclusions
but incomplete
explanations for
parts b and c
[1] at least one part
correct
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