Chapter 4 Congruence of Line Segments, Angles, and Triangles

Transcription

Name __________________________________________________________
Class ______________
Chapter 4-1 Postulates of Lines, Line Segments, and Angles
67
Date ______________
Section Quiz [20 points]
PART I
Answer all questions in this part. Each correct answer will receive 2 credits. No partial credit will be allowed. [12]
h
1. In the given figure, BC bisects ABD,
mABC 2x 5 and mCBD 3x 19. What is
mCBD?
3. In the given figure, AC > CE, B is the midpoint of
AC, and D is the midpoint of CE. If AC 6x 8 and
DE 5x 12, what is the length of AC?
C
A
C
D
B
A
B
(1) 24
(2) 48
D
✔ (3) 53
(4) 106
E
(1) 8
(3) 38
✔ (4) 56
(2) 28
g
4. Given: ABE
h
2. Which of the following statements a–f are not always
true?
a. Two points are contained in exactly one line.
b. If R, S, and T, in that order, are collinear, then
RS ST RT.
c. The bisector of a line segment passes through
exactly one point on that segment.
If m1 5x 2, BD bisects CBE, and
m2 3x 3, find mCBE.
C
D
d. If TM > MA, then M is the midpoint of TA.
1
e. If A, B, and C, in that order, are collinear and
AB 12AC, then B is the midpoint of A.
A
f. If AB bisects GH, then GH bisects AB.
(1) 16
(1) a and b
(3) d and e
(2) 51
(2) c and d
✔ (4) d and f
Copyright © Amsco School Publications, Inc.
2
B
E
(3) 78
✔ (4) 102
68
Name __________________________________________________________
5. Using the given figure, what is the degree measure
of x?
Class ______________
6.
Date ______________
C
B
(5x 2)°
g
D
25°
x°
30°
(3x 7)°
m
A
E
h
In the given figure, if BD bisects CBE and
(1) 65
✔ (2) 85
(3) 90
(4) 150
mDBE mBEA, then the measure of CBE is
✔ (1) 41
(3) 82
(2) 49
(4) 98
PART II
Answer all questions in this part. Each correct answer will receive 4 credits. Clearly indicate the necessary steps,
including appropriate formula substitutions, graphs, charts, etc. For all questions in this part, a correct numerical
answer with no work shown will receive only 1 credit. [8]
7. Fill in the missing Statements and Reasons.
Given: ABC, PQR, AB > PQ, and BC > QR
Prove: AC > PR
A
B
C
P
Q
R
Proof:
Statements
Reasons
1. ABC and PQR
1. Given.
2. AB > PQ and BC > QR
2. Given.
3. AB 1 BC > PQ 1 QR
3. Addition postulate.
4. AC 5 AB 1 BC, PR 5 PQ 1 QR
4. Partition postulate.
5. AC > PR
5. Substitution postulate.
Name __________________________________________________________
h
8. Given: CD bisects BCE, 2 3, and 4 1
B
1
Proof:
A
Reasons
h
1. CD bisects BCE.
1. Given.
2. 4 3
2. Definition of bisector.
3. 2 3 and 4 1
3. Given.
4. 1 2
4. Transitive property.
D
2
Prove: 1 2
Statements
Class ______________
34
C
E
69
Date ______________
70
Name __________________________________________________________
Class ______________
Chapter 4-2 Using Postulates and Definitions in Proofs
Date ______________
PART I
1. In the given figure, if m1 5x 4 and
m2 3x 48, what is m1?
D
(1) 17
(3) 29
✔ (4) 43
(2) 21.5
h
g
5. In the given figure, RT ' AB, mART 3x y,
and mTRB 2x y.
1 2
B
A
(1) 9
C
T
(3) 99
✔ (2) 81
(4) 126
2. In the given figure, m1 m2 m3. If
m1 7x 11 and m3 12x 4, what is
mABD?
E
D
1
B
C
2
3
A
(1) 32
✔ (3) 64
(2) 33
(4) 66
A
Which ordered pair represents the values (x, y)?
✔ (1) (36, 18)
(3) (45, 45)
(2) (18, 36)
(4) (90, 90)
6. In the given figure, AFB is a right angle. If
mAFE 3y 3, and mCFB 3y 3, what is
the value of y?
B
C
3. Segment BD is the perpendicular bisector of ABC. If
AB 3x 30 and BC 2x 10, what is the length
of AC?
(1) 20
(3) 40
(2) 30
✔ (4) 60
4. In the given figure, AB and EF are perpendicular
bisectors of each other at point M. If AM EM,
AB 4x 7, and EF 7x 20, what is the length
of AB?
E
A
M
B
R
A
D
F
E
✔ (1) 15
(3) 42
(2) 30
(4) 48
B
F
Name __________________________________________________________
Class ______________
Date ______________
PART II
A
7. Given: m1 m2
B
Prove: AEC BED
1
E
2
C
D
Proof:
Statements
Reasons
1. m1 m2
1. Given.
2. mBEC mBEC
2. Reflexive property of equality.
3. m1 mBEC m2 mBEC
4. mAEC m1 mBEC,
mBED m2 mBEC
5. mAEC mBED
6. AEC mBED
6. Definition of congruent angles.
8. Given: AF > CD
B
D
C
Prove: AC > FD
F
A
E
Proof:
Statements
Reasons
1. AF > CD
1. Given.
2. FC > FC
2. Reflexive property of congruence.
3. AF 1 FC > FC 1 CD
4. AC 5 AF 1 FC, FD 5 FC 1 CD
5. AC > FD
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72
Name __________________________________________________________
Chapter 4-3 Proving Theorems about Angles
Class ______________
Date ______________
PART I
1. In the given figure, if ACD and BCE intersect at C,
which statement must be true?
4. If PQR and AQB intersect at Q, then AQP and
AQR are
(1) congruent vertical angles
B
D
(2) congruent adjacent angles
(3) supplementary vertical angles
C
✔ (4) supplementary adjacent angles.
E
A
5. Which of the following statements is false?
(1) B, C, and D are collinear.
(2) ACE and ECD are complementary.
(3) ACE and BCD are supplementary.
✔ (4) ACB is congruent to DCE.
2. PQA and AQR form a linear pair of angles.
Which of the following statements is false?
(1) AQP and AQR are supplementary angles.
(2) 180 mPQA mAQR
h
h
h
h
✔ (3) QP and QA are opposite rays.
(4) QP and QR are opposite rays.
(1) An acute angle and an obtuse angle cannot be
vertical angles.
✔ (2) An acute angle and an obtuse angle cannot be
adjacent.
(3) Any two intersecting lines form adjacent
angles.
(4) Two angles that are adjacent cannot also be
vertical angles.
6. The number of degrees in a pair of supplementary
angles is represented by (2x 40) and (3x 10).
Find the number of degrees in the smaller of the two
angles.
(1) 20
3. The difference between the supplement and the
complement of an acute angle is
(1) 30
(2) 60
(2) 30
(3) 42
✔ (4) 44
✔ (3) 90
(4) 150
73
Name __________________________________________________________
Class ______________
Date ______________
PART II
7. a. Angles A and B are complementary. If the measure of A is 24 degrees more than twice the measure of B,
find the measures of A and B.
Answer: mA 68, mB 22
Solution:
Let mB x.
Then, mA 2x 24.
x 1 2x 1 24 5 90
3x 5 66
x 5 22
mB 22 and mA 2(22) 24 68.
b. Two angles, A and B, form a linear pair of angles. If the degree measure of A is 15 less than three times
that of B, find the measure of each angle.
Answer: mA 131.25, mB 48.75
Solution:
Let mB x.
Then, mA 3x 15.
x 1 3x 2 15 5 180
4x 5 195
x 5 48.75
mB 48.75 and mA 3(48.75) 15 131.25
C
M
8. Using the given figure, write a two-column proof for the following.
g
g
F
g
Given: AB intersects EF at P and CD at R, m1 m2
Prove: 3 and 4 are supplementary.
1
A
Proof:
3 P
E
2
4
Statements
Reasons
1. m1 m2
2. 2 and 4 form a linear pair.
3. 2 and 4 are supplementary.
4. m2 m4 180
5. 1 3
6. m1 m3
7. m3 m4 180
8. 3 and 4 are supplementary.
1. Given.
2. Definition of linear pair.
3. If two angles form a linear pair, then they are supplementary.
4. Definition of supplementary angles.
5. Vertical angles are congruent.
6. Definition of congruent angles.
B
R
D
74
Name __________________________________________________________
Class ______________
Chapter 4-4 Congruent Polygons and Corresponding Parts
Date ______________
PART I
In 1–4, use the figure below.
D
C
4. Which of the following is sometimes true?
H
G
(1) BC > FG
(2) EH > AD
✔ (3) FE > GH
A
B
E
F
Given: ABCD EFGH
1. Which of the following is always true?
(4) GH > CD
5. If ABC XYZ and XYZ DEF, then which
of the following is always true?
✔ (1) A E
(1) /A > /E
(2) A G
(2) AB > DF
(3) B H
(4) B E
2. Which of the following is always true?
(1) AB > GH
✔ (2) AB > EF
(3) AD > FG
(4) AD > GH
3. Which of the following is sometimes true?
(1) C G
✔ (3) C F
(4) CA > EF
6. Which of the following is not a property of
congruence?
(1) Any geometric figure is congruent to itself.
(2) Geometric figures congruent to the same
geometric figure are congruent to each other.
✔ (3) Angles formed by congruent sides are always
congruent.
(4) A given statement of congruence can always be
reversed.
✔ (2) E B
(3) F B
(4) H D
Name __________________________________________________________
Class ______________
Date ______________
PART II
7. In the given pair of congruent figures, name
C
B
a. all the congruent angles.
A
Answer: A F, B E, C H, D G
F
b. all the congruent sides.
D
G
E
H
Answer: AB > FE, BC > EH, CD > GH, AD > FG
8. In the given pair of congruent triangles, name
E
a. all the congruent angles.
Answer: A D, 2 4, 1 3
b. all the congruent sides.
Answer: AB > ED, BC > EF, CA > DF
2
D
C
13
F
A
4
B
75
76
Name __________________________________________________________
Class ______________
Chapter 4-5 Proving Triangles Congruent Using Side, Angle, Side
Chapter 4-6 Proving Triangles Congruent Using Angle, Side, Angle
Chapter 4-7 Proving Triangles Congruent Using Side, Side, Side
Date ______________
Version A [20 points]
PART I
1. In the given figure, BC > DF and mB mF.
C
B
A
3. Given: AB > CD and ABC BCD
E
D
C
F
D
B
A
What additional information would be needed to
prove the triangles congruent by ASA?
Which of the following could be used to prove
ABC DCB?
(1) mA mE
(2) mA mD
(1) ASA
✔ (3) mC mD
(2) AAA
✔ (3) SAS
(4) mC mE
2. In the given figure, B D, and E is the midpoint
of BD.
B
C
(4) SSS
4.
B
P
E
A
A
D
Which postulate justifies BCE DAE?
(1) SSS
(2) SAS
✔ (3) ASA
C
T
M
If AC 5(x 3) and TM x 7, then the two given
triangles are congruent when the value of x is
(1) 2
(2) 1
(3) 2.5
✔ (4) 5.5
(4) SSA
Name __________________________________________________________
5.
Class ______________
Date ______________
6. Given: BAC and EDF, C F
B
B
A
D
E
C
F
E
If ABC DEF, which of the following is not a
statement of congruence for these triangles?
(1) BCA EFD
A
C
F
D
Which additional piece of information is not
sufficient to prove BAC EDF?
✔ (2) ABC EDF
(3) ACB DFE
(1) A D and AC > DF
(4) CAB FDE
(2) B E and BC > EF
✔ (3) BC > EF and AB > DE
(4) BC > EF and AC > DF
PART II
B
7. Given: ABC is isosceles with vertex B. D is the midpoint of AC.
Prove: ABD CBD
Proof:
A
Statements
Reasons
1. ABC is isosceles with vertex B.
1. Given.
2. AB > CB
2. Definition of isosceles triangle.
3. D is the midpoint of AC.
3. Given.
4. AD > CD
4. Definition of midpoint.
5. BD > BD
6. ABD CBD
6. SSS.
D
77
C
78
Name __________________________________________________________
Class ______________
8. Given: AB bisects CBD and CAE DAE.
Date ______________
C
Prove: CAB DAB
E
A
3
4
1
2
B
D
Proof:
Statements
Reasons
1. AB bisects CBD.
1. Given.
2. 3 4
3. CAE DAE
3. Given.
4. CAE and 1 form a linear pair.
DAE and 2 form a linear pair.
5. CAE and 1 are supplementary.
5. If two angles form a linear pair, then they are supplementary.
DAE and 2 are supplementary.
6. 1 2
6. If two angles are congruent, then their supplements are congruent.
7. AB > AB
8. CAB DAB
8. ASA.
Name __________________________________________________________
Class ______________
Date ______________
Version B [20 points]
PART I
3. In the given figure, diagonal AT bisects A and
T. Which postulate can be used to prove that
ABT AKT?
1. In the given figure, DBE bisects ABC.
D
B
B
C
A
T
A
K
E
prove ABE CBD by ASA?
✔ (1) A C
(3) EB > BD
(2) E D
(4) AE > CD
✔ (3) ASA
(1) SSA
(2) SAS
(4) SSS
4. Given: PEN and ART, TR > NE
N
T
E
R
2. To prove ABC JKP by SAS, which additional
pairs of corresponding parts must also be congruent?
J
A
P
A
Which additional piece of information is not
sufficient to prove PEN ART?
(1) PN > AT and PE > AR
✔ (2) N T and PE > AR
B
C
P
K
(1) AC and JP
(3) AC and PK
(2) C and P
✔ (4) BC and KP
79
(3) N T and E R
(4) N T and PN > AT
80
Name __________________________________________________________
5.
B
Class ______________
Date ______________
6. Given: PQ > BC, Q C
3(16x 2)
B
P
C
A
D
6(5x 8)
The two triangular halves of the quadrilateral ABCD
are congruent by means of SAS if the value of x is
(1) 2
(3) 4
✔ (2) 3
(4) 6
R
Q
A
C
To prove PQR BCA by means of ASA, which
additional pair of corresponding parts must be
congruent?
(1) PR and BA
✔ (3) P and B
(2) R and A
(4) QR and CA
PART II
P
7. Given: PM ' AQ, and M is the midpoint of AQ.
Prove: APM QPM
Proof:
A
M
Q
Statements
Reasons
1. PM ' AQ
1. Given.
2. PMA and PMQ are right angles.
2. Definition of perpendicular lines.
3. PMA PMQ
3. Right angles are congruent.
4. M is the midpoint of AQ.
4. Given.
5. AM > QM
5. Definition of midpoint.
6. PM > PM
7. APM QPM
7. SAS.
Name __________________________________________________________
Class ______________
Date ______________
B
8. Given: ACDF, AD > CF, A F, 1 2
Prove: ABC FED
A
1
C
D
2
Statements
F
E
Proof:
Reasons
1. AD > CF
1. Given.
2. CD > CD
3. AD 2 CD > CF 2 CD
3. Subtraction postulate.
4. AD 5 AC 1 CD, CF 5 CD 1 DF
5. (AC 1 CD) 2 CD > (CD 1 DF) 2 CD
or AC > DF
6. A F, 1 2
6. Given.
7. 1 and ACB form a linear pair.
2 and FDE form a linear pair.
8. 1 and ACB are supplementary.
2 and FDE are supplementary.
9. ACB FDE
10. ABC FED
9. If two angles are congruent, then their supplements are
congruent.
10. ASA.
81
82
Name __________________________________________________________
Class ______________
Date ______________
Version C [20 points]
PART I
1. In the given figure, AC and BD intersect at E and
B D.
C
B
E
3. In triangles AED and BCD, AD > BD and
ED > CD. To prove these triangles congruent by
SAS, what additional parts need to be proven
congruent?
A
B
D
A
prove BAE DCE?
✔ (1) BE > ED
(3) A C
(2) AE > ED
E
(4) AC > BD
(1) E C
2. In the given figure, A P and AC > PR.
(2) A B
C
✔ (3) ADE BDC
(4) ADE BCD
Q
A
B
C
D
R
P
4. For which pair of triangles is the information
insufficient to prove that the triangles are congruent
using the SAS postulate?
(1)
(3)
(2)
(4)
✔
Which statement is sufficient to prove
ABC PQR?
✔ (1) C R
(2) C Q
(3) AC y PQ
(4) CB > QR
Name __________________________________________________________
5.
C
Date ______________
6. In the given figure, BD bisects B. Which
information would not be sufficient to prove that
ADB is congruent to CDB?
E
D
Class ______________
3(2x 5)
B
8(x 3)
B
A
The two given triangles, CBA and ADE, are
congruent when the value of x is
A
(1) 19.5
(1) AB > BC
(2) 6
(2) ADB CDB
(3) 1
D
C
✔ (3) BD bisects AC.
✔ (4) 4.5
(4) BD ' AC
PART II
B
7. Given: DB ' AB, AC ' DC, BE > CE
Proof:
C
E
Prove: ABE DCE
A
Statements
Reasons
1. DB ' AB, AC ' DC
1. Given.
2. ABE and DCE are right angles.
3. ABE DCE
4. BE > CE
4. Given.
5. BEA CED
6. ABE DCE
6. ASA.
83
D
84
Name __________________________________________________________
Class ______________
8. Given: ACDE, BA ' AE, FE ' AE, AB > EF and AD > CE
Date ______________
B
Prove: BAC FED
D
A
E
C
Proof:
Statements
Reasons
1. BA ' AE, FE ' AE
1. Given.
2. BAC and FED are right angles.
3. BAC FED
4. AB > EF and AD > CE
4. Given.
5. CD > CD
6. AD 2 CD > CE 2 CD
6. Subtraction postulate.
7. AD 5 AC 1 CD, CE 5 CD 1 DE
8. (AC 1 CD) 2 CD > (CD 1 DE) 2 CD
F
or AC > DE
9. BAC FED
9. SAS.
Name __________________________________________________________
Class ______________
Chapter 4 Congruence of Lines Segments, Angles, and Triangles
85
Date ______________
Chapter Review [40 points]
PART I
1. Which method cannot be used to prove triangles
congruent?
4. The given figure shows the intersection of three lines.
What is the value of x?
(1) SAS
✔ (2) SSA
2x
(3) AAS
x
(4) ASA
2.
3x
E
(1) 15
(3) 60
✔ (2) 30
(4) 90
5.
D
A
S
F
ADE AFE by reason of
(1) SAS
(2) ASA
(3) SSS
✔ (4) all of these
3. If an angle has a measure of 3x 10, then its
complement has a measure of
(1) 80 3x
✔ (2) 100 3x
(3) 3x 100
(4) 190 3x
P
Q
R
In the given figure, if mSQR 2x 5y and
mPQS x 2y, then x y =
(1) 30
✔ (2) 60
(3) 90
(4) 120
6. Which of the following conditions is false when
ABC 4x 8?
(1) The measure of the complement of ABC is 40
and x 10.5.
(2) The measure of the supplement of ABC is 140
and x 8.
(3) ABC is a right angle and x 20.5.
✔ (4) ABC is an acute angle and x 20.5.
86
Name __________________________________________________________
7. Identify the missing reasons in the proof of the
following.
Given: m1 ma 180,
m2 mb 180,
m1 m2
1
a
b
Class ______________
Date ______________
(1) Partition postulate, Subtraction postulate,
Substitution postulate
(2) Symmetric property, Transitive property,
Partition postulate
✔ (3) Transitive property, Substitution postulate,
Subtraction postulate
2
Prove: ma mb
(4) Partition postulate, Substitution postulate,
Symmetric property
8. In the given figure, if AN ' NC, DN ' NB,
mCNB x, mDNC 2x, and mAND y,
which statement is true?
Proof:
Statements
Reasons
1. m1 ma 180,
m2 mb 180,
m1 m2
2. m1 ma
m2 mb
3. m2 ma
m2 mb
4. ma mb
1. Given.
D
A
y
2.
3.
N
✔ (1) y x
(2) 2x y 90
2x
x
C
B
(3) 2y 2x x
(4) y x 30
4.
Name __________________________________________________________
Class ______________
Date ______________
PART II
9. In the given figure, MTRCD. If MT CD 9, TR RC, and MD 28,
what is the value of RD?
M
Answer: RD 14
Solution:
Let TR RC x.
MT 1 TR 1 RC 1 CD 5 MD
RC 1 CD 5 RD
9 1 x 1 x 1 9 5 28
5 1 9 5 14
2x 5 10
x55
10. What is the measure of angle X whose complement is one-third its supplement?
Answer: mX 45
Solution:
Let the complement of X be A.
Then, mX mA 90 and mX 3mA 180.
Solve for mA: mA 90 mX
Use substitution to solve for mX:
mX 3(90 mX) 180
2mX 270 180
90 2mX
45 mX
T
87
R
C
D
88
Name __________________________________________________________
Class ______________
Date ______________
PART III
11. A and B form a linear pair and mA mB. If mA 5x y and mB 4x y, what are the values of x
and y?
Answer: x 20, y 10
Solution:
5x 2 y 5 90
4(10) 1 y 5 90
4x 1 y 5 90
80 1 y 5 90
9x 5 90
y 5 10
x 5 10
12. Using the given figure:
C
D
a. Find the measures of EOB, DOC, and AOE.
Answer: mEOB 60, mDOC 30, mAOE 120
60°
O
A
B
b. Categorize each pair of angles:
E
(1) DOC and AOD
(2) EOB and AOD
(3) DOB and EOB
Answer: (1) Complementary angles, (2) Vertical angles, (3) Linear pair of angles
PART IV
13. In the following pairs of triangles, two parts of one triangle are given as congruent to two parts of the second
triangle. Name the third pair of parts that must be proved congruent in order to prove that the triangles are
congruent using SAS, ASA, or SSS. If more than one answer is possible, state all correct answers.
a.
B
A
E
C
D
F
Answer: AC > DF
89
Name __________________________________________________________
b.
H
Class ______________
Date ______________
L
G
J
K
R
B
M
Answer: G K or HJ > LM
c.
Q
P
C
A
Answer: P A or QR > BC
d.
B
A
D
C
Answer: ABD CBD or AD > CD
C
g
14. Given: AF and CG intersect at D, CB > GE, 1 2
Prove: CBD GED
A
1
B
E
2
3
D 4
F
G
Proof:
Statements
Reasons
1. CB > GE
1. Given.
2. 1 2
2. Given.
3. 1 and CBD form a linear pair.
2 and GED form a linear pair.
4. 1 and CBD are supplementary.
2 and GED are supplementary.
5. CBD GED
5. If two angles are congruent, then their supplements are congruent.
6. 3 4
7. CBD GED
7. ASA.
90
Name __________________________________________________________
Class ______________
Chapter 4 Congruence of Lines Segments, Angles, and Triangles
Date ______________
Cumulative Review [40 points]
PART I
1. Which of the following pairs of angles is neither
complementary nor supplementary?
(1) 78 and 12
(2) 57 and 123
(3) 37 and 143
✔ (4) 63 and 37
2. The statements “PQ QB” and “MB QB” are
true. Which of the following must also be true?
(1) PQ QB PB
(2) P, Q, and B are collinear.
(3) Q, B, and M are collinear.
✔ (4) PQ MB
✔ (2) isosceles
(1) L is a right angle.
✔ (2) M is a right angle.
(3) CE 5 LN
(4) FE ML
7. If the complement of P is greater then the
supplement of Q, which statement must be true?
(1) mP mQ 180
(2) mP mQ 90
3. If the measures of three angles of a triangle are given
as x 30, 4x 30, and 10x 30, then the triangle
must be
(1) obtuse
6. If CEF LMN and is E a right angle, then
which of the following is always true?
(3) right
(4) scalene
✔ (3) mP mQ
(4) mP mQ
8. For which pair of triangles is it not possible to prove
congruence?
(1)
4. The graph of the solution set of the equation
5(x 1) 6x is a
(1) line
(2) line segment
✔ (3) half-line
(4) ray
5. Let a represent “PQR is a right angle,” and b
represent “PQ ' QR.” Which of the following
represents the contrapositive of “If PQR is a right
angle, then PQ ' QR”?
(1) b → a
✔ (3) b → a
(2) b → a
(4) b → a
(2)
(3)
✔ (4)
Name __________________________________________________________
Class ______________
Date ______________
PART II
9. The degree measure of an angle is 24 more than twice the measure of its complement. Find the measure of the
angle and its complement.
Answer: angle 68°, complement 22°
Solution:
Let the measure of the complement x
Then the measure of the angle 2x 24
x 2x 24 90
2(22) 24 68
3x 66
x 22
10. The larger of two supplementary angles is 20° less than four times the smaller.
Find the measure of each angle.
Answer: 40° and 140°
Solution:
Let the measure of the smaller angle x
Then the measure of the larger angle 4x 20
x 4x 20 180
5x 200
x 40
4(40) 20 140
91
92
Name __________________________________________________________
Class ______________
Date ______________
PART III
11. ABC is a line segment. AB 5x 3, BC 4x 2, and AC 11x 7. Determine whether B is the midpoint of
ABC and justify your answer.
Answer: B is not the midpoint of ABC.
Explanation:
Since we are given ABC, AB BC AC.
Thus:
5x 3 4x 2 11x 7
9x 1 11x 7
6 2x
3x
Then AB 5(3) 3 12, and BC 4(3) 2 14.
Since AB BC, B is not the midpoint of ABC.
B
12. Given: BAD BCD, AC bisects BAD and BCD
Prove: m1 m2
A
Proof:
2
C
D
Statements
Reasons
1. BAD BCD
1. Given.
2. AC bisects BAD and BCD.
2. Given.
3. m1 12m/BAD, m2
4. 12m/BAD 5 12m/BCD
5. m1 m2
1
1
2m/BCD
4. Division postulate.
Name __________________________________________________________
Class ______________
Date ______________
PART IV
13. In the given figure, QM 8, MT 4y, RM x y, and MP 3x y. If
PR QT and MP MT, what is the value of QT?
Answer:
Q
R
QT 24
M
Solution:
PR 5 MP 1 RM
QT 5 QM 1 MT
5 x 1 y 1 3x 1 y
5 8 1 4y
P
T
5 4x 1 2y
Thus, 4x 1 2y 5 8 1 4y or 4 y 2x.
We also have that:
MP 5 MT
3x 1 y 5 4y
x5y
Therefore, 4 x 2x or x y 4, so QT 8 4(4) 24.
B
14. Given: AB > DC, BC ' DC, 1 is the complement of 2
Prove: ABC CDA
A
Proof:
Statements
1
2
3
D
Reasons
1. AB > DC, BC ' DC
1. Given.
2. BCD is a right angle.
2. Perpendicular lines form right angles.
3. mBCD 90
3. Definition of right angle.
4. mBCD m2 m3
5. m2 m3 90
5. Transitive property.
6. 2 is the complement of 3.
6. Definition of complement.
7. 1 is the complement of 2.
7. Given.
8. 1 3
8. Complements of congruent angles are congruent.
9. AC > AC
10. ABC CDA
10. SAS.
93
C

Chapter 4 Congruence of Line Segments, Angles, and Triangles

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