Download lg-ch4-2

Warm-Up Exercises
1. When are two angles congruent?
ANSWER
when they have the same measure
2. In ∆ABC, if m A = 64º and m
ANSWER
B = 71º, what is m
C?
45º
3. What property of angle congruence is illustrated by
this statement? If A
B and B
C, then
A
C.
ANSWER
Transitive Property
Warm-Up Exercises
Target
Proving triangles
congruent.
GOAL:
4.2 Identify congruent figures.
Warm-Up Exercises
Vocabulary
• congruent figures - figures the same shape and size;
two figures are congruent if and only if all corresponding
angles and all corresponding sides are congruent.
CPCTC - Corresponding Parts of Congruent Triangles
are Congruent
Third Angles Theorem 4.3 –
If two angles of one triangle are congruent to two angles
of another triangle, then the third angles are also congruent.
R-S-T Theorem for Congruent Triangles 4.4 –
Congruence of triangles is reflexive, symmetric, and
transitive.
Warm-Up1Exercises
EXAMPLE
Identify congruent parts
To write a congruence statement
for the triangles first identify all
pairs of congruent corresponding
parts.
SOLUTION
Corresponding angles
Corresponding sides
J
JK
The diagram indicates that
T,  K
TS, KL
JKL
S,
SR, LJ
L
RT
TSR.
congruence statement
R
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.
b.
You know that  F
m
F=m
Q
68 o = (6y + x)
68 = 6y + 8
10 = y
o
Q.
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-Up
Exercises
GUIDED
PRACTICE
for Examples 1, 2, and 3
In the diagram at the below, 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)
(b)
You know that H
(4x+ 5)° = 105°
4x = 100
x = 25
You know that H
m H m F =105°
F
F
CDEF.
H.
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
PROVE
AD
CB, DC
AB
ACD
CAB,
CAD
ACB
ACD
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°
5. By the definition of congruence, what additional
information is needed to know that
NDC
NSR.
ANSWER
DC
RS and DN
SN