Discovering Geometry An Investigative Approach X70 Workbook Ch04

User Manual: X70

Open the PDF directly: View PDF PDF.
Page Count: 8

24 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.1 • Triangle Sum Conjecture
Name Period Date
In Exercises 1–9, determine the angle measures.
1. p______, q______ 2. x______, y______ 3. a______, b______
4. r______, s______, 5. x______, y______ 6. y______
t______
7. s______ 8. m______ 9. mP______
10. Find the measure of QPT.11. Find the sum of the measures of
the marked angles.
12. Use the diagram to explain why 13. Use the diagram to explain why
Aand Bare complementary. mAmB mCmD.
A
B
E
D
C
A
CB
P
b
c
c
a
a
m
35⬚
s
76⬚
y
30 ⫹ 4x
100 ⫺ x
7x
85⬚x
31⬚
y
100⬚
s
t
r
79⬚
50⬚
23⬚ab
28⬚
17⬚
53⬚x
y
82⬚
98⬚
q
p
31⬚
Q
RS
P
T
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 24
Discovering Geometry Practice Your Skills CHAPTER 4 25
©2008 Key Curriculum Press
Lesson 4.2 • Properties of Isosceles Triangles
Name Period Date
In Exercises 1–3, find the angle measures.
1. mT______ 2. mG______ 3. x______
In Exercises 4–6, find the measures.
4. mA______, perimeter 5. The perimeter of LMO 6. The perimeter of QRS is
of ABC ______ is 536 m. LM ______, 344 cm. mQ______,
mM______ QR ______
7. a. Name the angle(s) congruent to DAB.
b. Name the angle(s) congruent to ADB.
c. What can you conclude about AD

and BC
?Why?
8. x_____, y_____ 9. PR QR and QS RS.10. Use the diagram to explain
If mRSQ 120°, what is why PQR is isosceles.
mQPR?
P
Q
RT
S
70⬚
55⬚
PRS
Q
4y
2x ⫹ y79⬚ ⫺ x
A
B
C
D
R
S
Qy
y ⫹ 31 cm
68⬚
M
O
Lx ⫹ 30⬚x
163 m
210 m
13 cm
a ⫹ 7 cm
A
a
B
C
39⬚
102⬚
x
110⬚
NG
A
T
R
I
58⬚
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 25
26 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.3 • Triangle Inequalities
Name Period Date
In Exercises 1 and 2, determine whether it is possible to draw a triangle
with sides of the given measures. If it is possible, write yes. If it is not
possible, write no and make a sketch demonstrating why it is not possible.
1. 16 cm, 30 cm, 45 cm 2. 9 km, 17 km, 28 km
3. If 17 and 36 are the lengths of two sides of a triangle, what is the range
of possible values for the length of the third side?
In Exercises 4–6, arrange the unknown measures in order from greatest
to least.
4. 5. 6.
7. x_____ 8. x_____ 9. What’s wrong with
this picture?
10. Explain why PQS is isosceles.
In Exercises 11 and 12, use a compass and straightedge to construct a
triangle with the given sides. If it is not possible, explain why not.
11. 12. Q
R
R
P
Q
P
B
C
C
A
B
A
x2x
PR
S
Q
120⬚
160⬚
C
A
B
x
158⬚
142⬚
66⬚
x
28⬚40⬚
71⬚
a
c
d
b
61⬚
32⬚
b
c
a
a
b
c
20
18
13
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 26
Discovering Geometry Practice Your Skills CHAPTER 4 27
©2008 Key Curriculum Press
Lesson 4.4 • Are There Congruence Shortcuts?
Name Period Date
In Exercises 1–3, name the conjecture that leads to each congruence.
1. PAT IMT 2. SID JAN 3. TS
bisects MA

,MT

AT
,
and MST AST
In Exercises 4–9, name a triangle congruent to the given triangle and
state the congruence conjecture. If you cannot show any triangles to be
congruent from the information given, write “cannot be determined” and
redraw the triangles so that they are clearly not congruent.
4. Mis the midpoint of AB
5. KITE is a kite with KI TI.6. ABC _____
and PQ
.
APM _____
KIE _____
7. MON _____ 8. SQR _____ 9. TOP _____
In Exercises 10–12, use a compass and a straightedge or patty paper and a
straightedge to construct a triangle with the given parts. Then, if possible,
construct a different (noncongruent) triangle with the same parts. If it is
not possible, explain why not.
10. 11. 12. XY
XZ
X
B
C
C
A
B
TS
U
T
S
U
y
x
G
D
O
T
P
2
4
6
8
10
246810
Q
T
U
R
S
T
N
O
M
T
I
E
K
P
M
Q
B
A
B
A
X
Z
C
Y
M
T
S
A
6
6
8
8
99
J
I
N
A
D
S
M
IA
T
P
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 27
28 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.5 • Are There Other Congruence Shortcuts?
Name Period Date
In Exercises 1–6, name a triangle congruent to the given triangle and state
the congruence conjecture. If you cannot show any triangles to be congruent
from the information given, write “cannot be determined” and explain why.
1. PIT _____ 2. XVW _____ 3. ECD _____
4. PS
is the angle bisector 5. ACN _____ 6. EFGH is a parallelogram.
of QPR.GQ EQ.
PQS _____ EQL _____
7. The perimeter of QRS is 350 cm. 8. The perimeter of TUV is 95 cm.
Is QRS MOL? Explain. Is TUV WXV? Explain.
In Exercises 9 and 10, construct a triangle with the given parts. Then, if
possible, construct a different (noncongruent) triangle with the same parts.
If it is not possible, explain why not.
9.
10.
A
AB
P
Q
PQ
x
40
x ⫹ 25
2x ⫺ 10
TU
V
X
W
LQ R
S
M
O
x
125
70
x ⫹ 55
2x ⫹ 15
EH
G
Q
K
L
F
P
N
A
RC
SP
Q
R
D
E
B
C
A
Z
Y
X
VW
PO
T
I
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 28
Discovering Geometry Practice Your Skills CHAPTER 4 29
©2008 Key Curriculum Press
Lesson 4.6 • Corresponding Parts of Congruent Triangles
Name Period Date
1. Give the shorthand name for each of the four triangle
congruence conjectures.
In Exercises 2–5, use the figure at right to explain why
each congruence is true. WXYZ is a parallelogram.
2. WXZ YZX 3. WZX YXZ
4. WZX YXZ 5. W Y
For Exercises 6 and 7, mark the figures with the given information. To
demonstrate whether the segments or the angles indicated are congruent,
determine that two triangles are congruent. Then state which conjecture
proves them congruent.
6. Mis the midpoint of WX

and 7. ABC is isosceles and CD

is the bisector
YZ
.Is YW

ZX
?Why? of the vertex angle. Is AD

BD
?Why?
In Exercises 8 and 9, use the figure at right to
write a paragraph proof for each statement.
8. DE
CF
9. EC
FD
10. TRAP is an isosceles trapezoid with TP RA and PTR ART.
Write a paragraph proof explaining why TA
RP
.
TR
A
P
DC
BA EF
C
BA D
Y
W
X
Z
M
Z
Y
WX
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 29
30 CHAPTER 4 Discovering Geometry Practice Your Skills
©2008 Key Curriculum Press
Lesson 4.7 • Flowchart Thinking
Name Period Date
Complete the flowchart for each proof.
1. Given: PQ
SR
and PQ
SR
Show: SP
QR

Flowchart Proof
2. Given: Kite KITE with KE
KI
Show: KT
bisects EKI and ETI
Flowchart Proof
3. Given: ABCD is a parallelogram
Show: A C
Flowchart Proof
ABCD is a parallelogram
_________________________
AB CD
______________
Definition of
___________
Same segment
_____________
_____________
____________
_____________
A
D
B
C
KET ______
______________
______________
______________
KITE is a kite
_______________
Definition
of bisect
KE KI
________________
ETK ITK
__________________
__________________
TK
I
E
PQS
______
Given
PQ SR
QS
______
SP QR
____________________________________
__________________
_________________ __________________
R
S
Q
P
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 30
Discovering Geometry Practice Your Skills CHAPTER 4 31
©2008 Key Curriculum Press
Lesson 4.8 • Proving Special Triangle Conjectures
Name Period Date
In Exercises 1–3, use the figure at right.
1. CD

is a median, perimeter ABC 60, and AC 22. AD _____
2. CD

is an angle bisector, and mA54°. mACD _____
3. CD

is an altitude, perimeter ABC 42, mACD 38°, and AD 8.
mB_____, CB _____
4. EQU is equilateral. 5. ANG is equiangular
mE_____ and perimeter ANG 51.
AN _____
6. ABC is equilateral, ACD is isosceles with base AC
,
perimeter ABC 66, and perimeter ACD 82.
Perimeter ABCD _____
7. Complete a flowchart proof for this conjecture: In an isosceles triangle,
the altitude from the vertex angle is the median to the base.
Given: Isosceles ABC with AC
BC
and altitude CD

Show: CD

is a median
Flowchart Proof
8. Write a flowchart proof for this conjecture: In an isosceles triangle, the
median to the base is also the angle bisector of the vertex angle.
Given: Isosceles ABC with AC
BC
and median CD

Show: CD

bisects ACB
AB
C
D
• •
•
A ________
ADC BDC
ADC and BDC
are right angles
__________________
____________________ Definition of altitude
Given
__________________
__________________
CD is an altitude
AC BC
AB
C
D
C
D
A
B
AB
C
D
DG4PS_893_04.qxd 11/1/06 10:20 AM Page 31

Navigation menu