Download congruent triangles

Name__
Class _
Test 3
Series 1
CONGRUENT TRIANGLES
Part I (10 points)
1 A triangle in which no two sides are
congruent is called a(n) __ triangle.
A
In the diagram, if ~-~ ---- ~,
then in order to prove ~ABC
~ ~EDC by HL, what additional
two sides must be congruent?
C
In a triangle, what name is given to a
line segment drawn from a vertex to the
midpoint of the opposite side?
4
2
3
H
If ~ ~ ~J, name
the base angles.
4
5 If ~ ------ ~J and
FO ~ FM, then
what property
justifies that ~
M
5
In problems 6-10, write A for always, S for
sometimes, or N for never.
triangle has 3 sides.
6
7 During the 5 minutes from 3:05 to 3:10, the
minute and hour hands always form
acute angles.
8 If a median of a triangle is also an altitude of
the triangle, the triangle is scalene.
9 If an angle is selected at random from a
triangle, the angle is obtuse.
10
9
If A -- (0, o), B = (10, 0),
and ~g is rotated 90° ~vith
respect to the origin, then
B will rotate to the point
(0, -- 10).
y
A (0, 0)
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x lO
B (~0, o)~ ~ o~
Test 3 [ 9
Date
Class
Name
CONGRUENT TRIANGLES
Test 3
Series 1
(continued)
Part II (20 points)
In problems 11-13, write a two-column proof.
Use a separate sheet of paper.
11
A
Given: Two triangles, ~ABC and
~ABD, standing on a desktop
called f
BC ~ BD
LABC ~- LABD
Prove:
F
12 Given: ~ = ~
G
H
KG ~ JF
Prove: LE=LH
13 Given: @O
LR~L_V
PO --- OW
Prove: TP- TW
R
V
\
PoW
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Name
Class__
CONGRUENT TRIANGLES
Date.
Test 3
Series 1
(continued)
Supply the missing reasons in the proof for
problem 14.
14
F
E
Given: LEBC ~LFCB
LABF ~- LDCE
Prove: ~EHC is isosceles.
Statements
1 !_EBC ~ LFCB
2 LABF is supp. to LFBC.
!_DCE is supp. to LECB..
3 LABF ~ LDCE
4 LFBC m/ECB
A
B
D
C
Reasons
1 Given
2 Iftwo angles forma straight angle,
then they are supplementary.
3 Given
4
5
6 ~FBC ~ ~ECB
7 FB~-EC
8 CH~FB
9 EC~CH
10 AEHC is isosceles.
6
7 CPCTC
8 Given
9
10
Part II][ (15 points)
15
16
The perimeter of ~BAG is 43.
AG=I6, AB=x +4,
BG = 2x + 2
By solving for x, determine
whether ~BAG is scalene,
isosceles, or equilateral.
A
2x+ 2
15
The circle has its center at the
origin and passes through (0, To the nearest hundredth, find 5).4
the area of the circle.
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Test 3 I 11
Date
Class__
Name _
CONGRUENT TRIANGLES
17
18
Test 3
Series 1
(continued)
EC=12, ET=3x-5,
VE= IO, ER=x +4
v
mLVEC=5x-2
mLRET = 3x + 10
On the basis of the given, what
c
must be the value of x?
Is AVEC ~ ARET?
R
T
17__
LB -~ L C
AB=3x +I, AC=2x +5,
BC=x +y
Solve for x.
If y < 2.97, then BC must be less
than what number?
19 ~RGH ~ ~ANE
GH = 10
mLG = 2w+ 2,
mLN = 17w - 658
By solving for w, tell
whether or not AN is
an altitude of ~ANE.
A
C
R
G
18__
A
HN
19
Part IV (5 points)
2O
RECT is a rectangle with
EC > RE. E = (4, 1) and C = (x, 1~,
If the area of RECT is 24 and all
sides of RECT have whole numbers
as lengths, find the 4 possible
coordinate values for x. (Hint: Find
all possible lengths for ~ first.)
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Name
Class
Date
Test 3
Series 3
CONGRUENT TRIANGLES
Part I (24 points)
B
c
In problems 1-6, choose the best answer.
I
If ABEC is isosceles, which of the
following must be given in order to
prove ABAE ~ ~CDE by ASA?
a L2~L3
b AB~CD
c AE~ED
d LABC~LDCB
1
Given AB ~ CD and AC ~ BD. In order
to prove the pair of overlapping triangles
congruent, what additional fact must
we use?
a Vertical angles are congruent.
b The reflexive property
c The addition property
d If A ,then
Given AE ~ ED and BE ~- CE, what
additional given is required in order
to ]Stove ~ABE ~ .&DCE by H.L.?
a LI~L4 b ABmCD
c L 1 and L4 are right angles, d BD ± AC
4
If ~PQR ~ ATWV, which of the
following is not necessarily true?
a LP~LT
b VT~-RP
c QR~WT
d LW~LQ
5 Complete the correspondence:
AGFP ~ __
a ~HPF
b ~HFL
c APFH
d ~HFP
H
6 Which of the following is
not necessarily true?
b APGL ~ ~PHL
a AGMF~- AGMH
d
c AKPF ~ AMFP
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Test 3 I 185
Date _
Class _
Name.
Test 3
Series 3
CONGRUENT TRIANGLES (continued)
In problems 7-12, write A for always, S for
sometimes, or N for never.
7 If ~-g ~ ~-~ in AABC, then LBAC ~ LABC
7
8 Two triangles are congruent if two sides
and an angle of one triangle are congruent
to two sides and an angle of the other.
.9 If three sides of one triangle are congruent
to three sides of another triangle, then each
pair of corresponding angles is congruent.
10 If APQR = AQRP, then ~PQR is equilateral.
11
If two altitudes of a triangle are congruent
then the medians to the same sides are
congruent.
12 An obtuse scalene triangle is congruent to
an acute isosceles triangle.
11_
12_
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Date
Class _
Name _
CONGRUENT TRIANGLES
Test 3
Series 3
(continued)
Part II (10 points)
In problems 13-17, give all answers accurate to
two decimal places.
13 Find the area of the shaded
region. (Use ~ ~ 3.1416.)
~
14 For what nonzero value of r, the radius,
is the circumference of a circle numerically
equal to its area?
14__
B
15 Given mLA is greater
than mLB. What are
the restrictions on the
value of x ?
x+ 10
15
A~ 2x+4
c
16 ~PQR ~ ASTV
pQ = x2
SV = 6
ST=x+6
TV = 3-x
Find all possible values for x.
Then find the perimeter of ~PQR.
16
17 The sides Of an isosceles triangle are whole
numbers. If the perimeter is 24, what is the
probability that the triangle is equilateral.
(Hint: list all valid possibilities first.)
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17
Test 3 1 187
Date _
Class __
Name
Test 3
Series 3
CONGRUENT TRIANGLES (continued)
(5, 11)
Part III (9 points)
W
~
18 The vertices of ~MRW are
(5, 11), (2, 4), and (9, 7)
respectively. Explain why
~MRW is isosceles.
(9, 7).
4)
X
18_
In problem 19, draw a diagram, state the given
and the conclusion, and write the proof.
19" If two points on the legs of an
isosceles h’iangle are equidistant
from the vertex, the segments joining
each to the endpoint of the base on
the opposite leg are congruent.
19_
Part IV (7 points)
¯
f
In problem 20, wr~te a paragraph proo.
2O Given: ~)P
BC ---- FE
AB ± FC
PA ~ PD
Prove: LC ~LF
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are congruent. 15 1 A--~ ~- ~ / Given 2 B is the midpoint of ~-~.
Given 3 D is the midpoint of i~7~. / Given 4 ~ ~ ~ / If two
segments are congruent, then their halves are congruent (Division
Property). 16 1 L 5 is supp. to/_ 6. / Given 2 /7 is supp. to/-6. /
Given 3 L 5 ~ L 7 / If two angles are supplementary to the same angle,
then they are congruent.
Part III (15 points)
17 72 18 21.76° or 21°45’36"
CT=5, EC= 10, RE = 5; area = 50
19 93 :~0 15
21 RT = 10,
Part IV (5 points)
22 1 L3iscomp. toLl./Given 2 L2iscomp. toL1./Given
3 L2 ~/-3 / If two angles are complementary to the same angle, then
they are congruent. 4 L4 ~ L5 / Given 5 LCOM ~- LPMO /
If congruent angles are added to congruent angles, then the sums are
congruent (Addition Property).
Chapter 3 Congruent Triangles
Part I
Part II
(10 points)
1 scalene 2 ~ and ~ 3 median
5 subtraction 6 A 7 A 8 N
4 LH and LJ
9 S 10 S
(20 points)
11 1 ~ ~ ~ / Given 2 LABC ~ !_ABD / Given
3 ~--~ ~ ~-~ / Reflexive Property 4 AABC ~- z~ABD / SAS (1, 2, 3)
5 ~-~/CPCTC 12 1 E~J~JJ/Given 2
Given 3 ~]~ ~- ~-~ / Given 4 ~ ~ ~ / Addition Property
5 ~EKG---~HJF/SSS(1,2,4) 6 LE~LH/CPCTC 13 1 ~)O/
Given
20~ ~ OV / All radii of a circle are congruent. 3 ~-~ ~ OW /
Given
4 ~VW/Subtraction Property 5 LR-~LV/Given
6 R-T~VT/If ~ ,then z~. 7 ARPT~AVWT/SAS(4,5,6)
8 TP ~ T~V / CPCTC 14 4 If two angles are supplementary to
congruent angles, then they are congruent.
5 Reflexive Property
6 ASA (1, 5, 4) 9 Transitive Property
10 If at least two sides of a
triangle are congruent, then the triangle is isosceles.
Part III (15 points)
15 isosceles 16 78.54
19 w =44;yes
17 x = 6; no
18 x =4;BC<6.97
Part IV (5 points)
~0 28, 16, 12, 10
Chapter 4 Lines in the Plane
Part I
(10 points)
1T2T3F4F
8 d 9 right
472 I Answers to Tests 3-4
5 T 6 L6 7 LB andLE~D
10 1.31
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11 mL~QG ff mLHAT + m/HOG / Substitution 12 mLBOG =
m/TAG / Substitution 13 LBOG ~- !TAG / Def. of congruence.
Z0 To have a complement, the supplement of the angle must have a
measure less than 90. If the supplement has measure less than 90, then the
angle itself must be greater than 90 and hence does not have a complement.
You cannot take the supplement of the complement because the angle does
not have a complement.
Chapter 3 Congruent Triangles
Part I
Part II
(24 points)
ld 2b 3c 4c
10 A 11 A 12 N
5d
6a 7S 8S 9A
(10 points)
1385.84
14 2
15 -2<x<6
16 x = -2;P=15
17~
Part III (9 points)
18 The two ft. ~s have legs 7 and 3, and are ~ by SAS. Thus ~ ~ ~, so
~ is isos.
A
19 Given: ~ABC is isos.
~
Prove: ~ ~ ~
B
C
A--~ ~ ~. ~ ~- ~, so ~-~ ~ ~ by subtraction. In an isos. ~, the base Ls
are ~, so LABC ~ LACB. By the Reflexive Property, ~ ~ ~. Then
~DBC ~ ZhECB by SAS..’. D--~ ~ ~ by C15CTC.
Part IV (7 points)
20 Since ~ 3_ ~-C and ~ 3_ ]TC, L BAC and L EDF are rt. L s. Then ~BAC
and ~EDF are ft. ~s. ~ ~- ~-~. Since ~ and ~-ff are radii of (~)P, ~C ~ ~.
P~ ~ P-~, so ~ m ~ by subtraction. Then ~ ~ ~ by addition.
~BAC ~ ~EDF by HL..’. LC ~ LF by CPCTC.
Chapter 4 Lines in the Plane
Part I (lO points)
1A2A
3N
92,3 102,5
4 S 5 A 6 A 71,5 82,4
Part II (~4 points)
¯ 11 ---~ 12 _3_4
13 (-18, 1)
14 ~ 15 (5, 7)
16 (-5, -7)
17 (-3,-2)
Part III (8 points)
o;
oo
18 x =25orx = -4; forx =25,150°, 30;forx = -4, 92,88
19 6 = 2 +2 10_,21 5 +~_~, so D is the midpoint of ~-C. Then ~ bisects
1 + 3 1. The slopes are
-5+3
AC, Or slope ~ = ~ = - 1, slope ~’~ = 6 -----~ =
negativ~ reciprocals, so ~ 3_ ]~’C. Therefore, since ~ bisects ~ and is ± to
~, then B~ is the 3_ bisector of AC.
:
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Answers to Tests 3-4 [ 497