Download CH 4 Review

NAME
_
DATE
Practice with Examples
For use with pages 194-201
Classify triangles by their sides and angles and find angie measures in
triangles
A triangle is a figure formed by three segments joining three noncollinear points.
An equilateral
triangle has three congruent sides.
An isosceles triangle has at least two congruent sides.
A scalene triangle has no congruent sides.
An acute triangle has three acute angles.
An equiangular
triangle has three congruent angles.
A right triangle has one right angle.
I An
obtuse triangle has one obtuse angle.
.
II
The three angles of a triangle are the interior angles.
I
When the sides of a triangle are extended, the angles that are adjacent to
the interior angles are exterior angles.
I
Theorem 4.1 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°.
I
I Theorem
4.2 Exterior Angle Theorem
,
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two nonadjacent interior angles.
Corollary to the Triangle Sum Theorem
The acute angles of a right triangle are complementary.
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Practice Workbook
Geometll"Y
vvit:h Examples
I
-,
LESSON
NAME
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DATE
I
Practice with Examples
CONTINUED
for use with pages 194-201
~
C/ass,_ifl.:;..y'_"n.;;.9_Tf:_iB_fl..::;:y_16_S
"""""""'
_
Classify the triangles by their sides and angles.
a.
b.
L
z
45°
5.7
4
J~";;";"----'-~
16
45°
XL.......J....----....:..::..~y
4
SOLUTiON
a. D.JKL has one obtuse angle and no congruent sides. It is an obtuse
scalene triangle.
b. LlXYZ has one right angle and two congruent sides. It is a fight
isosceles triangle .
.~~.'!!.~~~i!.~
..~~.~.
~l!.i!.I!!.I!.~'!..!
'"
.
Classify the triangie by its sides and angles.
1.
2.
5
3.
5.8
Geomet!'\f
Practice Workbook with Examples
18
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LESSON
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Practice IIvith Examples
CONTINUED
For use with pages 194-201
MUBIJ.6'~
Measures
b. Find the value of y.
a. Find the value of x.
(4x - 5)°
2VO
I
(3x+ 11)"
50°
f
I
I
a. From the Corollary to the Triangle Sum Theorem, you can write and
solve an equation to find the value of x.
The acute angles of a right triangle
are complementary.
I
x
"I
i
= 12
Solve for x.
b. You can apply the Exterior Angle Theorem to write and solve an
equation that will allow you to find the value of y.
90°
+ 50°
=
2yo
Apply the Exterior Angle Theorem.
y = 70
Solve for y.
!..;:.~!.~~~i!.~.!.~.~.~lJ.?l!!l!.~i!..?
Find the value of
4.
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.
x.
5.
Geometry
Practice Workbook with Examples
NAME
DATE
----------------------------------~------
Practice with Examples
For use with pages 202-210
identify congruent figures and corresponding parts
VOCABULARY
When two geometric figures are congruent, there is a correspondence:
between their angles and sides such that corresponding angles are
congruent and corresponding sides are congruent.
Theorem 4.3 Third Angles Theorem
If two angles of one triangle are congruent to two angles of another
triangle, then the third. angles are also congruent.
_
Using Properties of Congruent Figures
In the diagram, ABCDE
== FGHIJ
a. Find the value of x.
b. Find the value of y.
B
F
(3x+
4)
A
J
D
G
;
E
J
SOLUTiON
a. You know that AE
SoAE
10
=
=
3x
== Fl.
b. You know that LD
Fl.
So, mLD
+
4r
=
(8y - 9)°
56
=
8y
4
x=2
=
==
LI.
mLI.
y=7
)
Geometry
Practice Workbook with Examples
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LESSON
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Practice with Examples
CONTINUED
For use with pages 202-210
Exercises for Example 1
.......................................................................................................................................
In Exercises 1 and 2, for each pair of figures find (a) the value
of x and (0) the value of y.
1. MBC-=MEF
B
F
C
A
(5y - 3)°
2. ABCDEF
-= GHIJKL
0
G
(4x - 5)"
L
A
E
j'
).
F
K
I
I
Using the Third Angles Theorem
Find the value of x.
c
o
F
A .••....•..
~ __
f..LJ
E
B
SOLUTiON
In the diagram, LA = LD and LB = LE. From the Third Angles
Theorem, you know that LC
LF. So, mLC = mLF.
-=
From the Triangle Sum Theorem, mLC = 180° - 30° - 110° ~ 40°.
lI
mLC
=
mLF
40 = x
J!
.Third Angles Theorem
Substitute.
t
I
I
,
-!
I
)
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Geometry
Practice Workbook with Examples
LESSON
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CONTINUED
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DATE·
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:
""'."
Practice with Examples
For use with pages 202-210
.~~.~!.I?!.~f!.~.
!.~L
~1!.~I!!.I!.~f!.
~ ;
Find the value of x.
3.
4.
Geometry
Practice Workbook
with Examples
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NAME
_ DATE
Practice with Examples
For use with pages 212-219
.......- . -
=l
Prove that triangles are congruent using the SSS and SAS Congruence
Postulates
Postulate 19 Side-Side-Side (SSS) Congruence Postulate
. If three sides of one triangle are congruent to three sides of a second
triangle, then the two triangles are congruent.
.
I
• Postulate 20 Side-Angle-Side (SAS) Congruence Postulate
I.
.,EI»
~
If two sides and the included angle of one triangle are congruent to two
sides and the included angle of a second triangle, then the two triangles
are congruent.
.
Using the SAS Congruence Postulate
Prove that !:ABC
==
6DEF.
B
F
o
c
E
A
S«:U.UTRON
The marks on the diagram show that AB == DE, BC == EF, and
LB == LE. So, by the SAS Congruence Postulate, you know that
6ABC~
6DEF .
.~~~!.~~~~.~.
!.~.~
.~1!.~I!!p./f!..!
.
State the congruence postulate you would use to prove that
the two triangles are congruent.
1..
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Georne1:ry
Practice Workbook
with Examples
2
LIESSON
NAME
DATE ~----:-_---,
Practice with Examples
CONTINUED
For use with pages 212-219
~.'~-.
WJ<
~.
Congruent Triangles in a Coordinate Plane.
Use the SSS Congruence Postuiate to show
that MBC == 6CDE.
SOLUTION
.
Use the distance formula to show that corresponding sides are the same length. For all
lengths, d = .)(x2 - XI)2 + (h - Yl)2·
AB
= .)(-3- (-4))2 + (-3 - 2)2
= .)
CD =
12 + (- 5)2
= .J52 +
= .J26
=
=
. BC
.J( -1 - (- 3))2 + (0 - (- 3))2
= .J22+ 32
=
CA
=. DE,
= .J( -:-4
-' (-1))2
=
.J(-3)2 + 22
=
JI3
So, CA
=
(1 - 0)2
12 .
.J26
,--;
DE
= .J(1 - 4)2 + (3 - 1)2
=
and hence
+
=.J(-3)2+22
JI3
So, BC
(-1))2
AB == ro.
So, AB
=
CD, and hence
.J(4-
BE == DE.
+ (2 - 0)2
EC, and hence CA
ffi
EC = .)(-1 - 1)2
+ (0 -
3)2
= .J(-2)2 + (-3)2
= .JI3
== Ee.
So, by the SSS Congruence Postulate, you know that !::,ABC == 6CDE.
Geometry
Practice Workbook
vvith Examples
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LESSON
CONTINUED
NAME
~
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Practice with Examples
For use with pages 212-219
Exercise for .,,"Example ...................•....
2
.........
3. Prove that MBC == 6.DEF.
~"""""""""""
""
"" "" ""
,
"
"
"
"
"
"
". "
"" '"'' "'"
""
.
';
~
'
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Geometry
Practice Workbook with Examples,
9
t
I
NAME
DATE
~----------~----------------------
\
Practice with Examples
\
\
For use with pages 220-227
1
Prove that triangles are congruent using the ASA Congruence Postulate
and the AAS Congruence Theorem
.
,
,
I
t
1
Postulate 21 Angle-Side-Angle (ASA)'Congruence
Postulate
If two angles and the included side of one triangle are congruent to two
angles and the included side of a second triangle, then the two triangles
are congruent.
,
I Theorem
4.5 Angle-Angle-Side (AAS) Congruence Theorem
If two angles and a nonincluded side of one triangle are congruent to two
! angles and the corresponding nonincluded side of a second triangle, then
the two triangles are congruent.
I
_
Proving Triangles are Congruent Using the ASA Congruence Postulate
~
~
== EC, LB ==
!::ABC == 6DEC
Given: BC
Prove:
LE
=
8
Statements
Reasons
== Ee
1. Given
2. LB== LE
2.. Given
==
!::ABC ==
3. LACB
LDCE
3. Vertical Angles Theorem
4.
6DEC
4. ASA Congruence Postulate
Geometry
Practice Workbook
ms
o
SOLUTION
1. Be
em
with
Examples
i
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--~
LESSON
NAME
~
· D~E
Practice with Examples
-"""'::r,
CONTINUED
:..1;;..
',. _"--4f
For use with pages 220-227
Exercises lor Example t
..............................................................................................................................................
In Exercises 1 and 2, use the given information
the triangles are congruent.
1. Given: MC == AC
LNMC
to prove that
== DE, LA ==
Prove: 6BAE == 6CDE
2. Given: AE
and LBAC
are right angles.
Prove: 6NMC == 6BAC
N
o "'-'--------"
C
B
~
Proving Triangles ere Congruent Using AAS Congruence Theorem
!Sf
W~
~
.
Given: AD == AE, LB == L C
Prove: MBD
==
~wnr
lZi.IIlII&\!:WZ
::::a:SWt::::=i:i
:::eu!'!'JW~.=
c
B
MCE
A
Statements
Reasons
1. AD==AE
2. LB
3. LA
==
==
LC
2. Given
LA
3. Reflexive Property of Congruence
4. MBD==MCE
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.
1. Given
4. AAS Congruence Theorem
Geometry
Practice Workbook with Examples
LD
LESSON·
4.4
CONTINUED
NAME __'~~~~~~~~~~~~~~~~~~~~~
__
DATE ~
_
Practice with Examples
For use with pages 220-227
~~.f!!.~~~f!.~.
!.C!!.
.~l!.~I!!p/f!..?
.
In Exercises 3 and 4, use the given information
the triangles are congruent,
3. Given: LG
==
LB, CB II GA
to prove that
4. Given: LOMN
==
LONM,
LLMO== LINO
Prove: ·6GCA == 6BAC
Prove: 6MIN == 6NLM
'M
A"-----+--\"78
L
G
J
Geometry
Practice Workbook vvith Examples
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NAME
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Practice with Examples
For use with pages 229-235
Use congruent triangles to plan and write proofs
== PQ,
QS == RT
Given: PR
Prove:
SR
==
TQ
p
Plan for Proof: QS and RT are corresponding parts
of 6PQS and 6PRT and also of 6RQS and 6QRT.
The first set of triangles is easier to prove congruent
than the second set. Then use the fact that corresponding
parts of congruent triangles are congruent.
R
Q
SOU..!ITION
Statements
1. PR
. Reasons
== PQ
1. Given
2. PR = PQ
3. PR = PS
4. PQ
=
PT
6. SR
+ SR
== TQ
7. SR
=
5. PS
8.
PS
2. Definition of congruence
+ SR
+ TQ
=
PT
TQ
11. 6PQS
12. QS
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== RT
TQ
5. Subsitution
8. Subtraction property of equality
9. Definition of congruence
LP
==
+
7. Definition of congruence
= PT
==
4. Segment Addition Postulate
6. Given
9. PS== PT
10. LP
3. Segment Addition Postulate
10. Reflexive Property of Congruence
6PRT
11. SAS Congruence Postulate
12. Corresponding parts of congruent
triangles are congruent.
.
Practice Workbook
Geometry
with Examples
LESSON
NAME
_ DATE
Practice with Examples
CONTINUED
For use with pages 229-235
Exercises for Example 1
.......................................................................................................................................
.
.
Use the given information to prove the desired statement.
1. Given: RT==AS, RS==AT
Prove: L TSA
2. Given:
== LSTR
Prove:
Ll==L2==L3,L4==L5,ES==DT
HE
== HD
H
R
A
E
T
It>?}
o
,"J
s
\
~
,
Using More than One Pair of Triangles
Given: Ll
== L2, L5 == L6
B
Prove: AC.lBD
Plan for Proof: It can be helpful to reason backward
from what is to be proved. You can show that
AC.lBD if you can show L3 == L4. Notice that
L3 and L4 are corresponding parts of MBO and
MbO. You can prove MBO == bADO by SAS if
you first prove AB == AD. AB and AD are.
corresponding parts of MBC and MDC. You can
prove MBC == MDC by ASA.
A~----::------':-t-"---=---7C
o
./.;
lGeometry
Practice Workbook with Examples
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LESSON
NAME
_
DATE
Practice with Examples
CONTINUED
For use with pages 229-235
Statements
1. Ll ~ L2,L5
Reasons
~ L6
1. Given
2. Ae~AC
2. Reflexive Property of Congruence
3. MBe~MDe
3. ASA Congruence Postulate
4. AB~AD
4. Corresponding parts of congruent triangles are
congruent.
5. AO ~AO
5. Reflexive Property of Congruence
6. MBO==MDO
6. SAS Congruence Postulate .
7. L3 ~ L4
7. Corresponding parts of congruent triangles are
congruent.
8. If 2 lines intersect to form a linear pair of congruent
angles, then the lines are 1..
~;:.~!.'?/~~~.
t~.~
.~l!:?'!!l!.~~.?
.
In Exercises :3and 4, use the given information
desired statement.
to prove the
4. Given: LDAL ~ LBCM,
3. Given: PA ~ KA, LA ~ NA
Prove: AX ~ AY
LeDL~
LABM
DC~BA
N
Prove: AL == eM
rc+r+:
L
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-1-
-; C
B
Geometry
Practice Workbook with Examples
NAME __ ~
~
_
DATE
.Practice with Examples
For use with pages 236-242
Use properties
OIf
isosceles, equilateral, and right triangles
I
VOCABULARY
If an isosceles triangle has exactly two congruent sides, the two angles
adjacent to the base are base angles.
I
I
If an isosceles triangle has exactly two congruent sides, the angle opposite the base is the vertex angle.
I Theorem
I
I.
4.6 Base Angles Theorem
If two sides of a triangle are congruent, then the angles opposite them
are congruent
Theorem 4.7 Converse of the Base Angles Theorem
If two angles of a triangle are congruent, then the sides opposite them
I are congruent.
1i~
Corollary to Theorem 4.6
If a triangle is equilateral, then it is equiangular.
Corollary to Theorem 4.7
, If a triangle is equiangular, then it is equilateral.
Theorem 4.8 Hypotenuse-Leg (HL) Congruence Theorem
If the hypotenuse and a leg of a right triangle are congruent to the
hypotenuse and a leg of a second right triangle, then the two triangles
are congruent.
Using Properties.of Right Triangles
~
I~ -,
Given that LA and LD are Tight angles
and AB == DC, show that MBC == l',.DCB.
-
A~
~ __ ~C
SC)l.UTION
Paragraph proof
You are given thatLA and LD are
right angles. By definition, MBC and l',.DCB are right
triangles. You are also given that a leg of MBC, AB, is
congruent to a leg of ~DCB, DC. You know that the
B
D
hypotenuses of these two triangles, BC for both triangles, are congruent
because BC == BC by the Reflexive Property of Congruence. Thus, by
the Hypotenuse-Leg Congruence Theorem, MBC == /'::,DCB.
Geomet!'y
Practice Workbook with Examples
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LESSON
NAME
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DATE
Practice with Examples
l ~-
CONTINUED
For use with pages 236-242
.~~~!.~~~f!.~.
!.~.~.~lf.f!.I!!.e.~l!..
!
.
Write a paragraph proof.
1. Given: BCl.AD,
Prove:
AB
== DB
=
2. Given:mLIKL
MBC == L.DBC
IL
. !
Prove:
B
mLMLK
== MK
IK==ML
J
~
A
C
0
L
Using Equilateral and Is,!sceles.., T;iangles
M
. ,
Find the values of x and y.
SOLUTiON
Notice that tiABC is an equilateral triangle.
By the Corollary to Theorem 4.6, /':,.ABCis
also an equiangular triangle. Thus
mLA = mLABC = mLACB = 60°. So,
x = 60.
A
Notice also that f:..DBC is an isosceles triangle, and thus by the
Base Angles Theorem, mLDBC = mLDCB. Now, since
mLABC = mLABD + mLDBC, mLDBC = 60° - 30° = 300.
Thus, y = 30 by substitution.
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Geometry
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=
90°,
LESSON
NAME
CONTINUED
~
~----------------~--
_
DATE
Practice with Examples
«
For use with pages 236-242
~
Exercises for Example 2
-
........................................................................................................................................
Find the values of x and y.
3.
4.
20
5.
7
Geometry
Practice Workbook with Examples
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NAME
~
__ ~--------
DATE
Practice with Examples
For use with pages 243-250
Place geometric figures in a coordinate plane and write a coordinate proof
VOCABULARY
I
A coordinate proof involves placing geometric figures in a coordinate
I
plane and then using the Distance Formula and the Midpoint Formula,
I
as well as postulates and theorems, to prove statements about the figures. I
t
_
.,
Using the Distance formula
A right triangle has legs of 6 units and 8 units. Place the triangle in a
coordinate plane. Label the coordinates of the vertices and find the length
of the hypotenuse.
One possible placement of the triangle is shown.
Points on the same vertical segment have the
same x-coordinate, and points on the same horizontal
segment have the same y-coordinate.
y
I I
!
!
r
!
!
!
Distance Formula
1
Substitute.
:3
i
i
= -fiOo =
10
-
i
(O,O)!
I
!
i
I
!
!
I
I
I
!
1(8,0)
I I I I I
_
Use a coordinate plane and the Distance Formula to answer
the question.
'
1. A rectangle has sides of length 8 units and 2 units. What is the
length of one of the diagonals?
Copyright © McDougal Littell Inc.
I
I
-1
xl
i
Simplify and evaluate square root.
.~~.f!!.l?,~~~.~.
!.C!.~.~~c:.'!!P..~f!
..!
All rights reserved.
I
I i(8,6) I
I
I I Y i
!
i VI
I
i
dlA
I
1/
l/,
AI
!
= --1(8 - 0)2 + (6 - 0)2
,
/'
Use the Distance Formula to find d.
.\-
i I
I
!
Geometry
Practice Workbook vvith Examples
.
LESSON
NAME
_
DATE
CONTINUED
For use with pages 243-250
In the diagram, 6ABD
==
6CBD.
y
Find the coordinates of point B.
SOLUY§ON
. Because the triangles are congruent, AB == CB. So,
point B is the midpoint of AC: This means you can use
the Midpoint Formula to find the coordinates of point B.
) _ (Xl +
B(x, )) -
X')
2'
)'1 +
2
Y2)
~-(-72+ 0, ° 27' ) -_ (-22,22)
T
Midpoint
Formula
Substitute and Simplify .
.~;:.~!f?~~f!.~.!.l!.~.~l!.l!.i!!.I?~f!..?
2. In the diagram, 6FGH
coordinates of H.
Geometry
Practice Workbook with
Examples
==
.
6JIH. Find the
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LESSON
NAME
~
~
_
DATE
Practice with Examples
CONTINUED
for use with pages 243-250
WO'itillll1lI
a Coordinate
Write a plan to prove that 6,DEF == 6,DGF.
y
E(b,
c)
Given: Coordinates of figure DEFG.
Prove:
6DEF
==
6DGF
'F(a,O)
x
Plan for Proof Use the Distance Formula to
show that segments EF and GF have equal
lengths and that segments DE and DG have
equal lengths. Use the Reflexive Property of
Congruence to show that DF 2:: DF. Then use
SSS Congruence Postulate to conclude that
6DEF 2:: 6DGF.
Gib,
-c)
.~~.~~~~~l!.~.
!.~.~
.~l!.C?r!!.I!.~~.?
.
Describe a pian for the proof.
y
3. Given: Coordinates of figure ABeD.
Prove: MBC == 6CDA
A(O,O)
~~------~--------~~
x
D(2a, -b)
,I
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Geometry
Practice Workbook
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