Download Geo_Lesson 4_2

Geometry Lesson 4.2
Introduction to
Congruent Triangles
Warm-Up: Review of “Congruent”


Congruent segments have the same
length
_______________
Congruent angles have the same
measure
_______________
1. Definition of Congruent Figures

Geometric figures are congruent if
they have exactly the same size and
exactly the same shape
Congruent
NOT congruent
Congruence Implies Correspondence

Congruent figures have corresponding
angles and corresponding sides that are
congruent
B
A
ABC  DEF
C
A  D
Angles B  E
C  F
E
D
AB  DE
BC  EF
AC  DF
F
Sides
Get the Order Right!

CORRESPONDING PARTS MUST MATCH!
B
A
E
C
D
Can also write BCA  EFD, but
can NOT write ABC  EFD
F
Example 1: Naming Congruent Parts

Write a congruence statement and name
all corresponding angles and sides
Q
Congruence Statement
QRP  ACB
R
P
C
A
Q  A
QR  AC
R  C
PR  BC
P  B
PQ  AB
B
Practice 1: Naming Congruent Parts

Write a congruence statement and name
all corresponding angles and sides
_____  _____
_____  _____
_____  _____
_____  _____
_____  _____
_____  _____
_____  _____
Example 2: Finding Measures


In the figures below, NPLM  EFGH
Find the value of x and y
8
L
P
F
M
110°
G
(2x-3)
72°
10
N
E
(7y+9)°
H
1. What angle corresponds to E? N
7y + 9 = 72
y=9
2. What side corresponds to GH? LM
2x – 3 = 8
x = 5.5
Practice 2: Finding Measures


In the figures, ABCDEF  GHIJKL
Find values for x and y
2. Third Angles Theorem

If two angles of one triangle are congruent
to two angles of another triangle, then the
third angles are congruent
B
A
E
3rd 
C
D
If 2s , then 3rd s 
3rd 
F
Example 3: Third Angles Theorem

Find the value of x
Start by writing a congruence statement:
ABC  DEF
Then, apply the third angles theorem:
C  F mC = mF
From ABC, mC = 40° ( sum theorem)
mD = x° = 40°
Practice 3: Third Angles Theorem

Find x & mF using the third angles theorem
B
E
65°
A
55°
C
D
F
3. Proving Triangles are Congruent

Process: Use the definition of congruence
and the properties of the figure to develop
a logical argument
Example 4: Given the diagram, prove
PQR  NQM
N
R
92°
92°
P
Q
M
Example 4, cont.
N
R
92°
92°
Q
P
1. What do you need to know?
That all corresponding sides and angles are 
2. What do you already know?
All sides are  and mP = mN
3. What can you show?
PQR  NQM (vertical s) and
R  M (3rd s thm)
M
Example 4, cont.
N
R
92°
92°
Statement
P
Q
Reason
RP  MN, RQ  MQ, PQ  NQ
Given
mP = mN
Given
P  N
PQR  MQN
R  M
PQR  NQM
M
If =, then 
If vertical s, then 
If 2s , then 3rd s 
If corresp. s and
sides , then figures 
4. Congruent Triangle Properties
Reflexive
 Every triangle is congruent to itself
 Symmetric
 If ABC  DEF, then DEF  ABC
 Transitive
 If ABC  DEF and DEF  JKL, then
ABC  JKL

B
A
E
C
D
K
F
J
L
Closure


The design has only congruent triangles
If the total area is 96 ft², what is the area of
one triangle
32 congruent
triangles
96 ft² / 32 = 3 ft²
per triangle
Assignment

Ch 4.2 w/s