Download Chapter Summary and Review 5

Chapter
5
Summary and Review
Additional Resources
The following resources are available to help review the materials in
this chapter.
VOCABULARY
• corresponding parts, p. 233
• perpendicular bisector, p. 274
• congruent figures, p. 233
• distance from a point to a
line, p. 273
• proof, p. 243
• equidistant, p. 273
• line of symmetry, p. 284
Chapter 5 Resource Book
• Chapter Review Games and
Activities, p. 75
• Cumulative Review, Chs. 1–5
• reflection, p. 282
VOCABULARY REVIEW
Fill in the blank.
1. When two figures are __?__, their corresponding sides and their
corresponding angles are congruent. congruent
2. A(n) __?__ is a convincing argument that shows why a statement is
true. proof
3. If a point is on the __?__ of a segment, then it is equidistant from
the endpoints of the segment. perpendicular bisector
4. If a point is on the angle bisector of an angle, then it is __?__ from
the two sides of the angle. equidistant
5. A(n) __?__ is a transformation that creates a mirror image. reflection
5.1
Examples on
pp. 233–235
CONGRUENCE AND TRIANGLES
In the diagram, TABC c TRST. Identify all corresponding
congruent parts.
B
Corresponding angles
Corresponding sides
A
EXAMPLE
aA c aR
aB c aS
aC c aT
&* c RS
&*
AB
&* c RT
&*
AC
&* c ST
&*
BC
T
R
C
Use the triangles shown at the right to determine whether
the given angles or sides represent corresponding angles,
corresponding sides, or neither.
corresponding angles
&* and QR
&*
6. aJ and aR neither 7. KL
8. aK and aQ
corresponding sides
&* and LJ
& neither 10. JK
&* and PR
&* neither 11. aR and aL
L
9. PQ
corresponding angles
S
J
K
R
P
P
Chapter Summary and Review
291
291
Chapter Summary and Review continued
5.2
Examples on
pp. 241–244
PROVING TRIANGLES ARE CONGRUENT: SSS AND SAS
EXAMPLES
a.
Tell which congruence postulate you would use to show that
the triangles are congruent.
L
b.
A
B
G
C
H
P
M
N
F
a. In the triangles shown,
b. In the triangles shown,
&* c FG
&*,
S AB
&* c GH
&**, and
S BC
&* c FH
&**.
S AC
So, TABC c TFGH by the
SSS Congruence Postulate.
&** c NP
&*,
S LM
A aMLN c aPNL, and
&** c NL
&**.
S LN
So, TLMN c TNPL by the
SAS Congruence Postulate.
Decide whether enough information is given to show that the
triangles are congruent. If so, tell which congruence postulate you
would use.
12. E
B
13.
A
P
F
D
C
yes; SAS Congruence Postulate
5.3
a.
R
P
S
yes; SSS Congruence Postulate
P
yes; SAS Congruence Postulate
Examples on
pp. 250–253
Tell which congruence postulate or theorem you would use to show
that the triangles are congruent.
D
E
F
K
L
A aE c aK,
&* c KL
&*, and
S EF
A aF c aL.
So, TDEF c TJKL by the
ASA Congruence Postulate.
Chapter 5
S
b.
J
a. In the triangles shown,
292
R
T
PROVING TRIANGLES ARE CONGRUENT: ASA AND AAS
EXAMPLES
292
14. P
S
Congruent Triangles
U
T
Y
Z
X
b. In the triangles shown,
A aU c aZ,
A aT c aY, and
&* c XY
&*.
S ST
So, TSTU c TXYZ by the
AAS Congruence Theorem.
Chapter Summary and Review continued
Determine what information is needed to use the indicated postulate
or theorem to show that the triangles are congruent.
15. AAS Congruence
16. ASA Congruence
Theorem
T
17. ASA Congruence
Postulate
Postulate
B
U X
A
K
L
C
J
G
S
Z
Y
aS c aX
5.4
H
M
F
&* c FG
&*
AB
aJKM c aLMK
Examples on
pp. 257–259
HYPOTENUSE-LEG CONGRUENCE THEOREM: HL
Prove that TABC c TBFD.
EXAMPLE
&* ∏ AF
&*, DF
&** ∏ AF
&*
Given CB
&*.
B is the midpoint of AF
&* c BD
&*
AC
Prove TABC c TBFD
A
B
C
F
D
Show that the triangles are right triangles, the hypotenuses are
congruent, and that corresponding legs are congruent.
Statements
Reasons
&* ∏ AF
&*, DF
&** ∏ AF
&*, and B is
1. CB
1. Given
&*.
the midpoint of AF
2. aCBA and aDFB are right angles.
2. ∏ lines form right angles.
3. TABC and TBFD are right triangles.
3. Definition of right triangle
&* c BD
&*
H 4. AC
4. Given
&* c BF
&*
L 5. AB
5. Definition of midpoint
6. TABC c TBFD
6. HL Congruence Theorem
18. Use the information given in the diagram to fill in the missing
V
W
Z
Y
statements and reasons to prove that TUZV c T XYW.
U
Statements
Reasons
1. aUZV and aXYW are right angles.
Given
1. _________?_________
2. TUZV and TXYW are right
2. _________?_________
X
Def. of right T
triangles.
UV
&* c XW
&*
3. _________?_________
3. Given
&** c XY
&*
4. UZ
Given
4. _________?_________
a UZV c a XYW
5. _________?_________
5. HL Congruence Theorem
Chapter Summary and Review
293
293
Chapter Summary and Review continued
19–21. Check sketches.
19. HL Congruence Theorem
20. AAS Congruence Theorem
21. AAS Congruence Theorem
22. Statements (Reasons)
1. aC and aD are right angles.
(Given)
2. TABC and TBAD are right
triangles. (Definition of right
triangle)
& c BD
& (Given)
3. AC
& c AB
& (Reflexive Prop. of
4. AB
Congruence)
5. TABC c TBAD (HL
Congruence Theorem)
6. aCBA c aDAB (Corresp.
parts of c triangles are c.)
23. Statements (Reasons)
1. aJMN and aJKL are right
angles. (Given)
2. aJMN c aJKL (All right
angles are congruent.)
& c KL
& (Given)
3. MN
4. aJ c aJ (Reflexive Prop. of
Congruence)
5. TJMN c TJKL (AAS
Congruence Theorem)
& c JK
& (Corresp. parts of
6. JM
c triangles are c.)
24. Statements (Reasons)
1. aPQR and aTSR are right
angles. (Given)
2. aPQR c aTSR (All right
angles are congruent.)
& c TR
& (Given)
3. PR
4. aPRQ c aTRS (Vertical
Angles Theorem)
5. TPQR c TTSR (AAS
Congruence Theorem)
& c SR
& (Corresp. parts of
6. QR
c triangles are c.)
5.5
Examples on
pp. 265–267
USING CONGRUENT TRIANGLES
EXAMPLE
&** c KN
&* and aKML and aKNJ
In the diagram, KM
&* c KJ
&*.
are right angles. Prove that KL
L
K
M
J
First show that TMKL c TNKJ. Then use the fact that corresponding
&* c KJ
&.
parts of congruent triangles are congruent to show that KL
L
Sketch the triangles separately.
Mark the given information and any
other information you can conclude
from the diagram.
N
K
M
K
J
Statements
Reasons
&** c KN
&*
1. KM
1. Given
2. aKML and aKNJ are right angles.
2. Given
3. aKML c aKNJ
3. Right angles are congruent.
4. aK c aK
4. Reflexive Prop. of Congruence
5. TMKL c TNKJ
5. ASA Congruence Postulate
&* c KJ
&*
6. KL
6. Corresponding parts of congruent
triangles are congruent.
Sketch the overlapping triangles separately. Use the given
information to mark all congruences. Then tell what theorem or
postulate you can use to show that the triangles are congruent. 19–24. See margin.
&* c BD
&*
19. AC
C
&& c KL
&*
20. MN
D
&* c TR
&*
21. PR
M
L
J
A
B
P
P
K
N
22. Use the diagram and the information given in Exercise 19 above
to prove that aCBA c aDAB.
23. Use the diagram and the information given in Exercise 20 above
&* c JK
&*.
to prove that JM
24. Use the diagram and the information given in Exercise 21 above
&* c SR
&*.
to prove that QR
294
294
N
Chapter 5
Congruent Triangles
R
S
T
Chapter Summary and Review continued
5.6
Examples on
pp. 273–275
ANGLE BISECTORS AND PERPENDICULAR BISECTORS
EXAMPLE
Find AB.
C
By the Angle Bisector Theorem, AB DB.
A
2x 1 3x 2
Use Angle Bisector Theorem.
2x 3 3x
Add 2 to each side.
3x
31. Sample answer:
D
2x 1
3x 2
B
Subtract 2x from each side.
You are asked to find AB, not just the value of x.
ANSWER
AB 2x 1 2(3) 1 7
Use the diagram to find the indicated measure.
25. Find JM. 3
26. Find QR. 9
27. Find XY. 7
3
J
x4
3x 4
2x 1
W
L
P
K
5.7
X
P
M
S
Z
Examples on
pp. 282–285
REFLECTIONS AND SYMMETRY
EXAMPLE
Y
6x 1
R
Tell whether the red figure is a reflection of the
blue figure. Then determine the number of lines
of symmetry in the blue figure.
k
The red figure is a reflection of the blue figure in line k.
The blue figure has one line of symmetry.
Tell whether the red figure is a reflection of the blue figure.
Then determine the number of lines of symmetry in the blue figure.
28.
yes; 1
29.
k
no; 1
k
30.
k
yes; 1
31. Draw a six-sided figure that has two lines of symmetry. See margin.
Chapter Summary and Review
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295