Download lg_ch04_02 Apply Congruence and Triangles_teacher

4.2 Apply Congruence and
Triangles
• You will identify
congruent figures.
• Essential Question:
What are congruent
figures?
Tell students they will learn
how to answer this question
by studying the definition of
congruent figures
Warm-Up1Exercises
EXAMPLE
Identify congruent parts
Write a congruence statement for the
triangles. Identify all pairs of congruent
corresponding parts.
SOLUTION
The diagram indicates that
Corresponding angles
J
Corresponding sides JK
JKL
T,  K
TS, KL
TSR.
S,
SR, LJ
L
R
RT
Warm-Up2Exercises
EXAMPLE
Use properties of congruent figures
In the diagram, DEFG
SPQR.
a.
Find the value of x.
b.
Find the value of y.
SOLUTION
a.
You know that FG
FG = QR
12 = 2x – 4
16 = 2x
8=x
QR.
Warm-Up2Exercises
EXAMPLE
Use properties of congruent figures
b.
You know that  F
m
F=m
Q
68 o = (6y + x)
68 = 6y + 8
10 = y
Q.
o
Warm-Up3Exercises
EXAMPLE
Show that figures are congruent
PAINTING
If you divide the wall into
orange and blue sections
along JK , will the sections
of the wall be the same size
and shape?Explain.
SOLUTION
From the diagram, A
C and D
B because all
right angles are congruent. Also, by the Lines
Perpendicular to a Transversal Theorem, AB DC .
Warm-Up3Exercises
EXAMPLE
Show that figures are congruent
Then, 1
4 and 2
3 by the Alternate Interior
Angles Theorem. So, all pairs of corresponding angles
are congruent.
The diagram shows AJ CK , KD
JB , and DA BC .
By the Reflexive Property, JK KJ . All corresponding
parts are congruent, so AJKD
CKJB.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
CDEF.
1. Identify all pairs of congruent
corresponding parts.
SOLUTION
Corresponding sides:
Corresponding angles:
AB CD, BG DE,
GH FE, HA FC
A
G
C, B
E, H
D,
F.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
2. Find the value of x and find m
SOLUTION
(a)
You know that
(4x+ 5)° = 105°
4x = 100
x = 25
(b)
You know that H
m H m F =105°
H
F
F
CDEF.
H.
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 1, 2, and 3
In the diagram at the right, ABGH
3. Show that
PTS
CDEF.
RTQ.
SOLUTION
All of the corresponding parts of
PTS are congruent
to those of RTQ by the indicated markings, the
Vertical Angle Theorem and the Alternate Interior
Angle theorem.
Warm-Up4Exercises
EXAMPLE
Use the Third Angles Theorem
Find m
BDC.
SOLUTION
A
B and ADC
BCD, so by the Third
Angles Theorem, ACD
BDC. By the Triangle
Sum Theorem, m ACD = 180° – 45° – 30° = 105° .
ANSWER
So, m ACD = m BDC = 105° by the definition of
congruent angles.
Warm-Up5Exercises
EXAMPLE
Prove that triangles are congruent
Write a proof.
GIVEN
AD
CB, DC
ACD
PROVE
AB
CAB,
ACD
CAD
ACB
CAB
Plan for Proof
a. Use the Reflexive Property to show that
AC AC.
b. Use the Third Angles Theorem to show that
B
D
Warm-Up5Exercises
EXAMPLE
Prove that triangles are congruent
Plan in Action
STATEMENTS
REASONS
1.
AD
CB, DC
a. 2.
AC
AC.
BA
1. Given
2. Reflexive Property of
Congruence
3.
b. 4.
5.
ACD
CAD
B
ACD
CAB,
ACB
D
3. Given
4. Third Angles Theorem
CAB
5. Definition of
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 4 and 5
4. In the diagram, what is m
DCN.
SOLUTION
CDN
NSR, DNC
SNR then the third
angles are also congruent NRS
DCN = 75°
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 4 and 5
5. By the definition of congruence, what
additional information is needed to
know that
NDC
NSR.
ANSWER
DC
RS and DN
SN
Daily
Homework
Quiz
Warm-Up
Exercises
In the diagram,
1. m A =
?
ANSWER
2. FD
60°
?
ANSWER
3.
EDF
ANSWER
ABC
CA
?
BAC
DEF. Complete the statement.
Daily
Homework
Quiz
Warm-Up
Exercises
4. Write a congruence statement for the two small
triangles. Explain your reasoning.
ANSWER
WXZ
YXZ; The diagram tell us
that W
Y and WZX
YZX.
WXZ
YXZ by the Third s Thm.
From the diagram WX YX and
WZ YZ, and XZ XZ by Refl. Prop.
Of Segs.
• You will identify
congruent figures.
• Triangles can be proved
congruent by showing that
all 3 pairs of corresponding
sides and all 3 pairs of
corresponding angles
are congruent.
• If two angles of one triangle
are congruent to two angles
of another, then the third
angles are congruent.
• Essential Question:
What are congruent
figures?
Congruent figures are
figures that have exactly the
same size and shape.