Download Proving with SSS and SAS part 2

3 APPLY
ASSIGNMENT GUIDE
BASIC
Day 1: pp. 216–219 Exs. 6–21
Day 2: pp. 217–219 Exs. 22–28,
32–37, 39–46
AVERAGE
Day 1: pp. 216–219 Exs. 6–21
Day 2: pp. 217–219 Exs. 22–28,
32–37, 39–46
GUIDED PRACTICE
Vocabulary Check
✓
Concept Check
✓
2. The congruent angles
are not the angles
included between the
congruent sides.
Skill Check
ADVANCED
✓
Day 1: pp. 216–219 Exs. 6–21
Day 2: pp. 217–219 Exs. 22–28,
30–46
1. Sketch a triangle and label its vertices. Name two sides and the included
angle between the sides. See margin.
2. ERROR ANALYSIS Henry believes he can use the information given in the
diagram and the SAS Congruence Postulate to prove the two triangles are
congruent. Explain Henry’s mistake.
LOGICAL REASONING Decide whether enough information is given to
prove that the triangles are congruent. If there is enough information, tell
which congruence postulate you would use.
3. ¤ABC, ¤DEC
B
BLOCK SCHEDULE
E
H
D
yes; SAS Congruence
Postulate
K
yes; SSS Congruence
Postulate
PRACTICE AND APPLICATIONS
Extra Practice
to help you master
skills is on p. 809.
12. no
13. yes; SAS Congruence
Postulate
14. yes; SSS Congruence
Postulate
15. yes; SAS Congruence
Postulate
16. no
17. yes; SSS Congruence
Postulate
STUDENT HELP
S
R
J
no
STUDENT HELP
q
P
G
NAMING SIDES AND INCLUDED ANGLES Use the diagram. Name the
included angle between the pair of sides given.
Æ
Æ
Æ
Æ
Æ
Æ
6. JK and KL ™JKL
Æ
Æ
Æ
Æ
7. PK and LK
8. LP and LK ™KLP
HOMEWORK CHECK
To quickly check student understanding of key concepts, go over
the following exercises: Exs. 8, 16,
18, 20, 26, 28, 32, 34. See also the
Daily Homework Quiz:
• Blackline Master (Chapter 4
Resource Book, p. 55)
•
Transparency (p. 28)
5. ¤PQR, ¤SRQ
F
C
pp. 216–219 Exs. 6–21 (with 4.2)
pp. 216–219 Exs. 22–28, 30–37,
39–46 (with 4.4)
EXERCISE LEVELS
Level A: Easier
6–17
Level B: More Difficult
18–37
Level C: Most Difficult
38
4. ¤FGH, ¤JKH
A
9. JL and JK
Æ
10. KL and JL ™JLK
Æ
11. KP and PL
J
™LKP
L
™KJL
™KPL
K
P
LOGICAL REASONING Decide whether enough information is given to
prove that the triangles are congruent. If there is enough information, state
the congruence postulate you would use. 12–17. See margin.
12. ¤UVT, ¤WVT
13. ¤LMN, ¤TNM
L
T
14. ¤YZW, ¤YXW
T
Z
W
Y
W
V
M
U
N
X
HOMEWORK HELP
Example 1: Exs. 18,
20–28
Example 2: Exs. 19–28
Example 3: Exs. 12–17
Example 4: Exs. 20–28
Example 5: Exs. 30, 31
Example 6: Exs. 33–35
15. ¤ACB, ¤ECD
A
216
T
17. ¤GJH, ¤HLK
U
G
J
L
C
B
216
16. ¤RST, ¤WVU
D
Chapter 4 Congruent Triangles
E
R
S
V
W
H
M
K
DEVELOPING PROOF In Exercises 18 and 19, use the photo
Æ
Æ
Æ
Æ
of the Navajo rug. Assume that BC £ DE and AC £ CE .
COMMON ERROR
EXERCISE 2 Students may incorrectly say that two triangles can be
proven congruent using the SAS
Congruence Postulate when they
are given two pairs of corresponding congruent sides and the corresponding nonincluded angles.
Demonstrate that SSA is not a congruence postulate.
18. What other piece of information
is needed to prove that
¤ABC £ ¤CDE using the
Æ
Æ
SSS Congruence Postulate? AB £ CD
19. What other piece of information
is needed to prove that
¤ABC £ ¤CDE using the
SAS Congruence Postulate?
™ACB £ ™CED
20.
DEVELOPING PROOF Complete the proof by supplying the reasons.
Æ
Æ
Æ
Æ
GIVEN 䉴 EF £ GH ,
H
E
G
F
STUDENT HELP NOTES
Homework Help Students
can find help for Exs. 23–24 at
www.mcdougallittell.com.
The information can be printed
out for students who don’t have
access to the Internet.
FG £ HE
PROVE 䉴 ¤EFG £ ¤GHE
Statements
23. It
is given that
Æ Æ
˘
SP £TP and that PQ
bisects ™SPT. Then,
by the definition of
angle bisector,
™SPQ
£ ™TPQ.
Æ
Æ
PQ £ PQ by the
Reflexive Property of
Congruence, so
¤SPQ £ ¤TPQ by the
SAS Congruence
Postulate.
24. It
is given
that
Æ
Æ
£
RT
and
PT
Æ
Æ
QT £ ST .
™PTQ £ ™RTS by the
Vertical Angles
Theorem. Then
¤PQT £ ¤RST by the
SAS Congruence
Postulate.
INT
STUDENT HELP
NE
ER T
HOMEWORK HELP
Visit our Web site
www.mcdougallittell.com
for help with paragraph
proofs.
Æ
Æ
Æ
Æ
Æ
Æ
1. Sample answer:
Reasons
?㛭㛭㛭 Given
1. 㛭㛭㛭㛭㛭
?㛭㛭㛭 Given
2. 㛭㛭㛭㛭㛭
Property
?㛭㛭㛭 Reflexive
3. 㛭㛭㛭㛭㛭
of Congruence
?㛭㛭㛭 SSS Congruence Postulate
4. 㛭㛭㛭㛭㛭
1. EF £ GH
2. FG £ HE
3. GE £ GE
4. ¤EFG £ ¤GHE
TWO-COLUMN PROOF Write a two-column proof. 21, 22.See margin.
Æ
Æ
Æ
Æ
Æ
Æ
Æ
Æ Æ
Æ
21. GIVEN 䉴 NP £ QN £ RS £ TR,
22. GIVEN 䉴 AB £ CD, AB ∞ CD
PQ £ ST
PROVE 䉴 ¤ABC £ ¤CDA
PROVE 䉴 ¤NPQ £ ¤RST
N
S
A
T
D
1
Y
X
Z
åX is included between X苶Y苶 and
X苶Z苶, åY is included between
X苶Y苶 and Y苶Z苶, and åZ is included
between X苶Z苶 and Y苶Z苶.
21. Statements (Reasons)
1. N
苶P苶 • Q
苶N
苶 • R苶S苶 • T苶R苶, P苶Q
苶 • S苶T苶
(Given)
2. †NPQ • †RST
(SSS Congruence Postulate)
22. See Additional Answers beginning on page AA1.
2
q
P
R
B
C
PARAGRAPH PROOF Write a paragraph proof. 23, 24. See margin.
Æ
˘
23. GIVEN 䉴 PQ bisects ™SPT,
Æ
Æ
SP £ TP
PROVE 䉴 ¤SPQ £ ¤TPQ
Æ
Æ Æ
PROVE 䉴 ¤PQT £ ¤RST
q
P
P
S
T
T
q
Æ
24. GIVEN 䉴 PT £ RT, QT £ ST
S
R
4.3 Proving Triangles are Congruent: SSS and SAS
217
217
Software Help Instructions for
several software packages are
available in blackline format in the
Chapter 4 Resource Book, p. 44
and at www.mcdougallittell.com.
25. Statements (Reasons)
1. A
苶C苶 • B
苶C苶, M is the midpoint
of A
苶B
苶. (Given)
2. A
苶M
苶• B
苶M
苶 (Definition of
midpoint)
3. C苶M
苶 • C苶M
苶 (Reflexive Property
of Congruence)
4. †ACM • †BCM
(SSS Congruence Postulate)
26. Statements (Reasons)
1. B
苶C苶 • A
苶E苶, B
苶D
苶• A
苶D
苶, D
苶E苶 • D
苶C苶
(Given)
2. BD = AD, DE = DC (Definition of
congruent segments)
3. BD + DE = AD + DC (Addition
property of equality)
4. BD + DE = BE, AD + DC = AC
(Segment Addition Postulate)
5. BE = AC (Substitution property)
6. B
苶E苶 • A
苶C苶 (Definition of congruent segments)
7. A
苶B
苶• A
苶B
苶 (Reflexive Property
of Congruence)
8. †ABC • †BAE
(SSS Congruence Postulate)
33–35, 38, 46.
See Additional Answers beginning on page AA1.
27. Since
it
is given
that
Æ
Æ
Æ
PA
£ PB
£ PC and
Æ
Æ
AB £ BC , ¤PAB £
¤PBC by the SSS
Congruence Postulate.
For Lesson 4.3:
• Practice Levels A, B, and C
(Chapter 4 Resource Book, p. 45)
• Reteaching with Practice
(Chapter 4 Resource Book, p. 48)
•
See Lesson 4.3 of the
Personal Student Tutor
For more Mixed Review:
•
Search the Test and Practice
Generator for key words or
specific lessons.
218
Æ
STUDENT HELP
NE
ER T
Æ
Æ
Æ Æ
Æ
26. GIVEN BC £ AE, BD £ AD,
Æ
M is the midpoint of AB.
PROVE ¤ACM £ ¤BCM
Æ
DE £ DC
PROVE ¤ABC £ ¤BAE
C
B
C
D
A
M
B
Æ
Æ
Æ
Æ
E
A
Æ
27. GIVEN PA £ PB £ PC,
Æ
Æ Æ
Æ
Æ
Æ
28. GIVEN CR £ CS, QC fi CR,
AB £ BC
QC fi CS
PROVE ¤PAB £ ¤PBC
PROVE ¤QCR £ ¤QCS
q
P
C
C
R
B
A
S
TECHNOLOGY Use geometry software to draw a triangle. Draw a line
29.
and reflect the triangle across the line. Measure the sides and the angles
of the new triangle and tell whether it is congruent to the original one. See margin.
SOFTWARE HELP
Visit our Web site
www.mcdougallittell.com
to see instructions for
several software
applications.
Æ
25. GIVEN AC £ BC,
29. The new triangle and
the original triangle
are congruent.
Writing Explain how triangles are used in the object shown to make it
more stable.
30.
31.
30, 31. Sample answers
are given.
30. The cross pieces form
triangles which are
rigid, ensuring that the
supports keep their
shape.
31. The struts that go from
the body of the plane to
the wing form
triangles, making the
wing structure rigid.
ADDITIONAL PRACTICE
AND RETEACHING
PROOF Write a two-column proof or a paragraph proof. 25–28. See margin.
Æ
28. It
is given that
CR £
Æ
Æ
CS and that QC is
perpendicular to both
Æ
Æ
CR and CS . If two lines
are perpendicular, then
they intersect to form
four right angles, so
™QCR and ™QCS are
right angles. By the
Right Angle
Congruence Theorem,
™QCR £ ™QCS. By
the Reflexive Property
Æ
of
Congruence, QC £
Æ
QC . Then ¤QCR £
¤QCS by the SAS
Congruence Postulate.
INT
STUDENT HELP NOTES
32. Use the method
described in the
activity on page 213.
Note, however, that the
same compass setting
will be used to
construct the legs of
the triangle.
32.
CONSTRUCTION Draw an isosceles triangle with vertices A, B, and C.
Use a compass and straightedge to construct ¤DEF so that ¤DEF £ ¤ABC.
See margin.
xy USING ALGEBRA Use the Distance Formula and the SSS Congruence
Postulate to show that ¤ABC £ ¤DEF. 33–35. See margin.
33.
34.
y
C
E
D
y
y
E
D
D
C
A
E
2
1
A
B
F
5 x
C
Chapter 4 Congruent Triangles
F
1
B
1
218
35.
F
x
A
B
1
x
Test
Preparation
Æ
Æ Æ
Æ
36. MULTIPLE CHOICE In ¤RST and ¤ABC, RS £ AB, ST £ BC, and
Æ
Æ
TR £ CA. Which angle is congruent to ™T? C
A
¡
B
¡
™R
C
¡
™A
D
¡
™C
4 ASSESS
cannot be determined
37. MULTIPLE CHOICE In equilateral ¤DEF, a segment is drawn from point F
Æ
Transparency Available
Use the diagram. Name the
included angle between the
pairs of sides given.
to G, the midpoint of DE. Which of the statements below is not true? B
★ Challenge
A
¡
Æ
B
¡
Æ
DF £ EF
Æ
Æ
DG £ DF
C
¡
Æ
Æ
DG £ EG
D
¡
38. CHOOSING A METHOD Describe how
EXTRA CHALLENGE
www.mcdougallittell.com
¤DFG £ ¤EFG
y
to show that ¤PMO £ ¤PMN using the
SSS Congruence Postulate. Then find a way
to show that the triangles are congruent using
the SAS Congruence Postulate. You may not
use a protractor to measure any angles. Compare
the two methods. Which do you prefer? Why?
See margin.
N
D
F
1
O
1
x
P
SCIENCE CONNECTION Find an important angle in the photo. Copy the angle,
extend its sides, and use a protractor to measure it to the nearest degree.
(Review 1.4)
39.
40.
G
H
1. D
苶G
苶 and G
苶E苶 åDGE
2. E苶H
苶 and G
苶H
苶 åEHG
Using the diagram above, decide
whether enough information is
given to prove that the triangles
are congruent. If there is enough
information, state the congruence
postulate you would use.
3. 䉭DGF, 䉭EGH no
4. 䉭DGE, 䉭EGH yes;
SSS Congruence Postulate
EXTRA CHALLENGE NOTE
Challenge problems for
Lesson 4.3 are available in
blackline format in the Chapter 4
Resource Book, p. 52 and at
www.mcdougallittell.com.
USING PARALLEL LINES Find m™1 and m™2. Explain your reasoning.
(Review 3.3 for 4.4)
41.
42.
43.
1
1 129ⴗ
2
1
ADDITIONAL TEST
PREPARATION
1. WRITING Explain the difference
between the SSS Congruence
Postulate and the SAS
Congruence Postulate.
2
2
57ⴗ
LINE RELATIONSHIPS Find the slope of each line. Identify any parallel or
perpendicular lines. (Review 3.7) 44–46. See margin.
44.
45.
y
3
¯˘ 3
slope of BD = }};
5
¯˘ ¯˘
AC fi BD
¯˘
45. slope of EF = º2, slope of
¯˘
¯˘ ¯˘
GH = º2, EF ∞ GH
E
M
MIXED REVIEW
39, 40. Sample answers are
given.
39. The measure of each of
the angles formed by two
adjacent “spokes” is
about 60°.
40. The measure of each of
the angles formed by two
adjacent sides of a cell of
the honeycomb is about
120°.
41. m™2 = 57° (Vertical
Angles Theorem),
m™1 = 180° º m™2 =
123˚ (Consecutive Interior
Angles Theorem)
42. m™1 = 180° º 129° = 51°
(Linear Pair Postulate),
m™2 = m™1 = 51°
(Alternate Exterior Angles
Theorem)
43. m™1 = 90°
(Corresponding Angles
Postulate), m™2 = 90°
(Alternate Interior Angles
Theorem or Vertical
Angles Theorem)
¯˘
5
44. slope of AC = º }},
DAILY HOMEWORK QUIZ
A
E
D
1
x
C
y
G
1
1
2
B
46.
y
F
œ
x
H
R
P 1
1
4.3 Proving Triangles are Congruent: SSS and SAS
x
Sample answer: Both are used to
prove triangles congruent. SSS is
used when three sides of one triangle are congruent to three sides
of a second triangle. SAS is used
when two sides and the included
angle of one triangle are congruent to two sides and the included
angle of a second triangle.
219
219