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Warm-Up Exercises
Suppose that ∆XYZ
1. XY
ANSWER
2.
Z
ANSWER
?
RS
?
T
Lesson 4.6, For use with pages 256-263
∆RST. Complete each statement.
Warm-Up Exercises
Suppose that ∆XYZ
3. m
4. If A
find x.
ANSWER
∆RST. Complete each statement.
?
S=m
ANSWER
Lesson 4.6, For use with pages 256-263
Y
B, m
50
A = (2x + 40)º, and m
B = (3x – 10)º,
Warm-Up1Exercises
EXAMPLE
Use congruent triangles
Explain how you can use
the given information to
prove that the hanglider
parts are congruent.
GIVEN
PROVE
1
QT
2,  RTQ
RTS
ST
SOLUTION
If you can show that QRT
SRT, you will know that
QT ST. First, copy the diagram and mark the given
information.
Warm-Up1Exercises
EXAMPLE
Use congruent triangles
Then add the information that you can deduce. In this
case, RQT and RST are supplementary to congruent
angles, so  RQT
RST. Also, RT RT .
Mark given information.
Add deduced information.
Two angle pairs and a non-included side are congruent,
QRT
SRT
so by the AAS Congruence Theorem,
.
Because corresponding parts of congruent triangles are
ST.
congruent, QT
Warm-Up
Exercises
GUIDED
PRACTICE
1.
for Example 1
Explain how you can prove that
A
C.
SOLUTION
AB
BC
AD
DC
Given
Given
BD
BD
Reflexive property
ANSWER
Thus the triangle
ABD
BCD by SSS
Warm-Up2Exercises
EXAMPLE
Use congruent triangles for measurement
Surveying
Use the following method to
find the distance across a
river, from point N to point P.
•
Place a stake at K on the
near side so that NK NP
•
Find M, the midpoint of NK .
•
Locate the point L so that NK
are collinear.
KL and L, P, and M
Warm-Up2Exercises
EXAMPLE
Use congruent triangles for measurement
•
Explain how this plan allows you to find the distance.
SOLUTION
Because NK NP and NK KL , N
and K are congruent right angles.
Because M is the midpoint of NK , NM KM . The
vertical angles KML and NMP are congruent. So,
MLK
MPN by the ASA Congruence Postulate.
Then, because corresponding parts of congruent
triangles are congruent, KL
NP . So, you can find the
distance NP across the river by measuring KL .
Warm-Up3Exercises
EXAMPLE
Plan a proof involving pairs of triangles
Use the given information to write a
plan for proof.
GIVEN
1
PROVE
2,
BCE
3
4
DCE
SOLUTION
In
BCE and
DCE, you know 1
2 and CE
CE .
If you can show that CB CD , you can use the SAS
Congruence Postulate.
Warm-Up3Exercises
EXAMPLE
Plan a proof involving pairs of triangles
To prove that CB CD , you can first prove that
CBA
CDA. You are given 1
2 and 3
4.
CA CA by the Reflexive Property. You can use the
ASA Congruence Postulate to prove that CBA
CDA.
Plan for Proof
Use the ASA Congruence Postulate to prove that
CBA
CDA. Then state that CB CD . Use the SAS
Congruence Postulate to prove that BCE
DCE.
Warm-Up
Exercises
GUIDED
PRACTICE
2.
for Examples 2 and 3
In Example 2, does it matter how far from point N
you place a stake at point K ? Explain.
SOLUTION
No, it does not matter how far from point N you place
a stake at point K . Because M is the midpoint of NK
NM
MK
MNP
MKL are
both right triangles
KLM
NMP
MKL
MNP
Given
Definition of right
triangle
Vertical angle
ASA congruence
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
No matter how far apart the strikes at K and M are
placed the triangles will be congruent by ASA.
3.
Using the information in the
diagram at the right, write a plan to
prove that PTU
UQP.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 2 and 3
STATEMENTS
TU
PQ
Given
PT
QU
Given
PU
PU
Reflexive property
PTU
PTU
REASONS
UQP
SSS
UQP By SSS
This can be done by showing right triangles QSP and
TRU are congruent by HL leading to right triangles
USQ and PRT being congruent by HL which gives you
PT
UQ
Warm-Up4Exercises
EXAMPLE
Prove a construction
Write a proof to verify that the construction for copying
an angle is valid.
SOLUTION
Add BC and EF to the diagram. In the
construction, AB , DE , AC , and DF are
all determined by the same compass
setting, as are BC and EF . So, you can
assume the following as given
statements.
GIVEN
PROVE
AB
D
DE, AC
A
DF, BC
EF
Warm-Up4Exercises
EXAMPLE
Prove a construction
Show that CAB
FDE, so you can
conclude that the corresponding parts
and D are congruent.
Plan For
Proof
STATEMENTS
1.
Plan in
Action 2.
3.
AB
AC
REASONS
DE
DF, BC
FDE
D
A
CAB
A
1. Given
EF
2. SSS Congruence
Postulate
3. Corresp. parts of
are .
Warm-Up
Exercises
GUIDED
PRACTICE
4.
for Example 4
Look back at the construction of an angle bisector
in Explore 4 on page 34. What segments can you
assume are congruent?
SOLUTION
AC and AB