Download 4-5 Reading Strategies Use a Graphic Aid

Name
LESSON
4-5
Date
Class
Reading Strategies
Use a Graphic Aid
4WOTRIANGLES
YES
4HETRIANGLESARECONGRUENT
BY!NGLE3IDE!NGLE!3!
CONGRUENCE
!RETWOANGLESANDANONINCLUDEDSIDEOFONE YES
TRIANGLECONGRUENTTOTHECORRESPONDINGANGLES
ANDTHENONINCLUDEDSIDEOFTHEOTHERTRIANGLE
4HETRIANGLESARECONGRUENT
BY!NGLE!NGLE3IDE!!3
CONGRUENCE
!RETWOANGLESANDTHEINCLUDEDSIDEOFONE
TRIANGLECONGRUENTTOTWOANGLESANDTHE
INCLUDEDSIDEINTHEOTHERTRIANGLE
NO
NO
!RETHEHYPOTENUSEANDALEGOFARIGHT
TRIANGLECONGRUENTTOTHEHYPOTENUSEAND
ALEGOFANOTHERRIGHTTRIANGLE
4HETRIANGLESARECONGRUENT
BY(YPOTENUSE,EG(,
CONGRUENCE
YES
Use the flowchart to determine, if possible, whether the following pairs of
triangles are congruent. If congruent, write ASA, AAS, or HL—the postulate
you used to conclude that they are congruent. If it is not possible to
conclude that they are congruent, write no conclusion.
1.
2.
4
1
9
"
3
0
3.
2
6
3
5
5.
5
4
4.
7
"
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6.
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Copyright © by Holt, Rinehart and Winston.
All rights reserved.
42
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8
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,
Holt Geometry
LESSON
4-5
Review for Mastery
LESSON
4-5
Triangle Congruence: ASA, AAS, and HL continued
If two angles and a nonincluded side of one triangle are congruent to the corresponding
angles and nonincluded side of another triangle, then the triangles are congruent.
F
FH is a nonincluded
side of F and G.
H
G
K
Graph each set of lines. Then:
a. Identify two congruent triangles that are formed by the lines.
b. Write a paragraph proof to justify your statement in part a.
_
J
Congruent Triangles on the Coordinate Plane
When proving that two triangles are congruent, you may need to use your
knowledge of lines on the coordinate plane.
Angle-Angle-Side (AAS) Congruence Theorem
_
Challenge
JL is a nonincluded
side of J and K.
1. y 5
y 4
a. ABC
y x 1
y 3x 2
Y
EDC
b. Proofs may vary.
L
FGH JKL
"
!
#
Special theorems can be used to prove right triangles congruent.
X
Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of
another right triangle, then the triangles are congruent.
J
K
2. y 3
x 1
P
L
JKL MNP
5. Describe the corresponding parts and the justifications
for using them to prove the triangles congruent by AAS.
y 2x 1
y 2x 3
A
1
Prove: ABD CBD
D
2
0
B
Given: BD is the angle bisector of ADC.
3
3. y 4
x 2
a. JKM
yx2
y x
*
V
T
+
6. UVW WXU
4. On a separate sheet of paper, write equations for a different set of lines that
meet to form congruent triangles. Identify the congruent triangles, and write
a proof to justify the congruence.
7. TSR PQR
_ _
Yes; UV WX (Given) and
_ _
No; you need to know that
_
UW UW (Reflex. Prop. of )
_
Answers will vary.
TR PR.
39
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All rights reserved.
Holt Geometry
Problem Solving
Melanie is at hole 6 on a miniature golf course. She
walks east 7.5 meters to hole 7. She then faces south,
turns 67° west, and walks to hole 8. From hole 8, she
faces north, turns 35° west, and walks to hole 6.
4-5
4/12/07 12:22:30 PM
7.5 m
23°
35°
67°
8
1. Draw the section of the golf course described.
Label the measures of the angles in the triangle.
7
YES
4HETRIANGLESARECONGRUENT
BY!NGLE3IDE!NGLE!3!
CONGRUENCE
!RETWOANGLESANDANONINCLUDEDSIDEOFONE YES
TRIANGLECONGRUENTTOTHECORRESPONDINGANGLES
ANDTHENONINCLUDEDSIDEOFTHEOTHERTRIANGLE
4HETRIANGLESARECONGRUENT
BY!NGLE!NGLE3IDE!!3
CONGRUENCE
NO
Yes; the is uniquely determined by AAS.
NO
3. A section of the front of an English
in
_ Tudor
_ home
_ is shown
_
the diagram. If you know that KN LN and JN MN,
can you use HL to conclude that JKN MLN ? Explain.
No; you need to know that KJN and
+
,
!RETHEHYPOTENUSEANDALEGOFARIGHT
TRIANGLECONGRUENTTOTHEHYPOTENUSEAND
ALEGOFANOTHERRIGHTTRIANGLE
.
*
4HETRIANGLESARECONGRUENT
BY(YPOTENUSE,EG(,
CONGRUENCE
YES
-
Use the flowchart to determine, if possible, whether the following pairs of
triangles are congruent. If congruent, write ASA, AAS, or HL—the postulate
you used to conclude that they are congruent. If it is not possible to
conclude that they are congruent, write no conclusion.
LMN are rt. .
Use the diagram of a kite for Exercises 4 and 5.
_
AE is the angle bisector of DAF and DEF.
1.
#
Reading Strategies
Use a Graphic Aid
!RETWOANGLESANDTHEINCLUDEDSIDEOFONE
TRIANGLECONGRUENTTOTWOANGLESANDTHE
INCLUDEDSIDEINTHEOTHERTRIANGLE
67°
2. Is there enough information given to determine the location of holes
6, 7, and 8? Explain.
"
Holt Geometry
4WOTRIANGLES
Possible drawing:
6
40
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
LESSON
Triangle Congruence: ASA, AAS, and HL
Use the following information for Exercises 1 and 2.
X
Q
R
,
-
W
S
4-5
P
X
X
Y
LKM or JKM MKL
b. Proofs may vary.
Determine whether you can use the HL Congruence Theorem to
prove the triangles congruent. If yes, explain. If not, tell what else
you need to know.
LESSON
_
BD BD (Reflex. Prop. of )
001-082_Go08an_CRF_c04.indd 39
C
A C (Given), ADB CDB (Def. of bisector),
U
%
Y
a. PQS RQS
b. Proofs may vary.
_
_
$
M
N
2.
4
1
9
"
3
$
0
%
2
5
&
!
#
ASA
8
:
no conclusion
'
3.
(
!
4. What can you conclude about
DEA and FEA?
4
5.
B DEA FEA by AAA.
G BCA HGA by AAS.
H BCA HGA by ASA.
D DEA FEA by SAS.
J It cannot be shown using the given
information that BCA HGA.
41
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1
%
0
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no conclusion
F BCA HGA by HL.
C DEA FEA by ASA.
4.
7
8
5. Based on the diagram, what can you
conclude about BCA and HGA?
A DEA FEA by HL.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
6
3
5
!
"
#
HL
6.
%
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Copyright © by Holt, Rinehart and Winston.
All rights reserved.
76
+
*
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&
AAS
Holt Geometry
(
,
ASA
42
Holt Geometry
Holt Geometry