~
Proof
Real-World
Connection
History According to legend,
one of Napoleon's officers used
congruent triangles to estimate
the width of a river. On the
riverbank, the officer stood up
straight and lowered the visor
of his cap until the farthest thing
he could see was the edge of the
opposite bank. He then turned
and noted the spot on his side
of the river that was in line with
his eye and the tip of his visor.
Given: LDEG and LDEF are right angles; LEDG
== LEDF.
The officer then paced off the distance to this spot and declared that distance to be
the width of the river! Use congruenttriangles to prove that he was correct.
Prove: EF
== EG
Reasons
Statements
1.
2.
3.
4.
5.
6.
<'iff Quick(heck
8
LEDG == LEDF
DE == DE
LDEG and LDEF are right angles.
LDEG == LDEF
LDEF == LDEG
EF == EG
21
For more exercises, see Extra SkiJI, Word Problem, and Proof Practice.
Practice by Example
1. Developing Proof State why the two triangles
are congruent. Give the congruence statement.
Then list all the other parts of the triangles that
are congruent by CPCTe.
Example 1
(page 221)
for
• Help
Given
Reflexive Property of Congruence
Given
All right angles are congruent.
ASA Postulate
CPCTC
About how wide was the river if the officer stepped off 20 paces and each pace
was about
ft long?
EXERCISES
e
1.
2.
3.
4.
5.
6.
---
== LCBD,
LBDA == LBDC
Prove: AB == CB
Proof 2. Given: LABD
== -ER,
ME==RO
3. Given: OM
-
Prove:
LM
== LR
o
B
c
A
R
M
D
222
Chapter 4
Congruent Triangles
E
4. Developing Proof Two cars of the same
Example 2
model have hood braces that are identical,
connect to the body of the car in the same
place, and fit into the same slot in the hood.
(page 222)
= VE,AR = EH,RC = HV
Given: CA
Complete the proof that the hood braces
hold the hoods open at the same angle.
Prove: LARC
o
=
LEHV
Proof: It is given that the three sides of the triangles are congruent,
so.6ARC
.6EHVbya.~.Thus,LARC
LEHVbyb.~.
=
Apply Your Skills
~
=
5. Earth Science Some distances are best measured indirectly.
Sinkhole SwallowsH ouse
The large sinkhole in this photo occurred
suddenly in 1981 in Winter Park, Florida,
following a severe drought. Increased
water consumption lowers the water
table. Sinkholes form when caverns in
the underlying limestone dry up and
collapse.
A geometry class indirectly measured the
distance across a sinkhole. The distances
they measured are shown in the diagram.
Explain how to use their measurements to
find the distance across the sinkhole.
---
=
Proof 6. Given: LSPT
SP
Prove: LS
5
= OP
=
= Y P, LC = LR,
7. Given: YT
LOPT,
LT=
Prove: CT
L0
LP
= RP
o
T
C
y
p
K
Copy and mark the figure to show the given information.
Explain how you would use SSS, SAS, ASA, or AAS
with CPCTC to prove LP 8;! LQ.
,'n1ine
Homework Video Tutor
Visit: PHSchool.com
8. Given: P K
Web Code: aue-0404
= QK,
p
T
KL bisects LPKQ.
9. Given: KL is the perpendicular bisector of PQ.
10. Given:
~
IT
..L PQ, KL
11. Given: LQPS
Prove: PQ
= LRSP,
= SR
-
p
bisects LPKQ.
LQ
Q
= LR
5
Lesson 4-4
Using Congruent Triangles: CPCTC
223
A
U. Wrtting Karen cut this pattern for the stained.
glass shown here so that AB = CB and AD = CD.
Must LA be congruent to LC? Explain.
B
D
c
13. Constructions The construction of a line perpendicular
to line through point P on is shown here.
a. Which lengths or distances are equal by
construction?
~
b. Explain why you can conclude that CP
is perpendicular
to e. (Hint: Do the
A
construction. Then draw CA and CB.)
e
e
c
p
B
14. Error Analysis The proof is incorrect. Find the error and tell how you would
correct the proof.
B
Given: LA :::= LC, BD bisects LABC.
For a guide to' solving
Exercise 14, see page 226.
Prove: AB
.i
CB
:::=
Reasons
Statements
1. LA:::= LC
1. Given
2. BD bisects LABC
2. Given
3. Ll
3. Definition
of bisect
4. Definition
of bisect
L2
:::=
4. AD:::= CD
5. MBD:::=
Proof 15. Given: BA
:::=
D
C
5. AAS Theorem
6CBD
6. AB:::= CB
---
A
6. CPCTC
16. Given:
BC,
e
.1 AB,
P is on
BD bisects LABC.
Prove: BD .1 AC, BD bisects AC.
Prove:
PA
e bisects
AB at C,
e.
= PB
p
B
A
A
~
D
e
C
C
B
17. Constructions In the construction of the bisector of LA below, AB :::= AC
because they are radii of the same circle. BX :::= CX because both arcs had the
same compass setting. Tell why you can conclude that AX bisects LBAC
--
B
Problem Solving Hint
In the third diagram,
what two triangles
must be congruent,
and why?
---
A
C
Proof 18. Given: BE 1. AC, DF 1. AC,
-BE:::= DF,AF:::= EC
-
Prove: AB
B
:::=
Chapter 4
Congruent
Triangles
Prove:
-
JK II QP,JK "" QP
KQ bisects IP.
DC
A!1-?Q
224
19. Given:
C
Q
Challenge ~Proof 20. Given: P R II M G, M p II G R
Prove: Each diagonal of PRGM divides
PRGM into two congruent triangles.
~Proof zi, Given: PR II MG,MP II GR
Prove: PR == MG,MP == GR
(Hint: See Exercise 20.)
Multiple Choice·
Pr-------.------,R
G
M
22. In the diagram, """RXW == LJXT. Which
statement is NOT necessarily true?
== LR
C. WX == JX
A. LJ
B. L W
D. RW
T
R
== L T
== JT
w
J
23. Which is true by (P(Te?
F. AC bisects BD
H. LABE
==
LEDC
B
==
G. LBAC
-
J. BC
==
LDCA
-
DC
A~C
24. Which is not true by (peTe?
==
DE
C. LBCA
==
A. BE
Short Response
."".,. for
Help
Lesson 4-3
B. LBAC
LDCE
D. AB
==
LDAC
== AD
D
"""ABC== LADC
Exercises 23-24
25. In the diagram, KB bisects L VKT and KV == KT.
a. What do you need to show in order to
conclude LKBV == LKBT? State whether it is
possible to show this and justify your answer.
b. Show that VB == TB.
I
K
V~T
B
What postulate or theorem can you use to prove the triangles congment?
°67
27
Lesson 2-5
28. The measure of an angle is 10 more than the measure of its supplement. Find
the measures of both angles.
Lesson 2-3
If possible, use the Law of Detachment
draw a conclusion, write not possible.
to draw a conclusion. If it is not possible to
29. If two nonverticallines are parallel, then their slopes are equal. Line m is
nonvertical and parallel to line n.
30. If a convex polygon is a quadrilateral, then the sum of its angle measures is 360.
Convex polygon ABCD E has five sides.
31. If a quadrilateral is a square, then it has four congruent sides. Quadrilateral
. ~-. ABCD has four congruent sides.
enIine
lesson quiz, PHSchool.com, Web Code: aua·0404
Lesson 4-4
Using Congruent
Triangles: CPCTC
225