Geometry Lesson 4.2
Introduction to
Congruent Triangles
October 23, 2007
Objectives
� Students will be able to:
� Define: congruent figures
� Identify congruent figures and their corresponding sides and angles
� Find angle measures in congruent figures
� Find the measure of a 3rd angle given two congruent angles in a triangle
� Prove two triangles are congruent
Warm-Up: Review of “Congruent”
� Congruent segments have the same
_______________
� Congruent angles have the same
_______________
length
measure
1. Definition of Congruent Figures
� Geometric figures are congruentcongruent if
they have exactly the samesame sizesize and
exactly the samesame shapeshape
NOT congruentCongruent
Congruence Implies Correspondence
� Congruent figures have corresponding
anglesangles and corresponding sidessides that are
congruent
A
B
C D
E
F
∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆DEF
∠∠∠∠A ≅≅≅≅ ∠∠∠∠D AB ≅≅≅≅ DE∠∠∠∠B ≅≅≅≅ ∠∠∠∠E BC ≅≅≅≅ EF∠∠∠∠C ≅≅≅≅ ∠∠∠∠F AC ≅≅≅≅ DF
Angles Sides
Get the Order Right!
� CORRESPONDING PARTS MUST MATCH!
Can also write ∆∆∆∆BCA ≅≅≅≅ ∆∆∆∆EFD, but
can NOT write ∆∆∆∆ABC ≅≅≅≅ ∆∆∆∆EFD
A
B
C D
E
F
Example 1: Naming Congruent Parts
� Write a congruence statement and name
all corresponding angles and sides
P
Q
R
Congruence Statement∆∆∆∆QRP ≅≅≅≅ ∆∆∆∆ACB
A
C
B
∠∠∠∠Q ≅≅≅≅ ∠∠∠∠A QR ≅≅≅≅ AC
∠∠∠∠R ≅≅≅≅ ∠∠∠∠C PR ≅≅≅≅ BC
∠∠∠∠P ≅≅≅≅ ∠∠∠∠B PQ ≅≅≅≅ AB
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
P
L
M
N72°
110°
8
10E
FG
H(7y+9)°
(2x-3)
1. What angle corresponds to ∠∠∠∠E?
2. What side corresponds to GH?
∠∠∠∠N
LM
7y + 9 = 72 y = 9
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 anglesthird angles are congruent
A
B
C D
E
F
If 2∠∠∠∠s ≅≅≅≅, then 3rd ∠∠∠∠s ≅≅≅≅
3rd ∠∠∠∠ 3rd ∠∠∠∠
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
A
B
C D
E
F
(2x+30)°55°
65°
3. Proving Triangles are Congruent
� Process: Use the definition of congruence
and the properties of the figure to develop
a logical argumentlogical argument
92°
92°Q
M
N
R
P
Example 4: Given the diagram, prove
∆∆∆∆PQR ≅≅≅≅ ∆∆∆∆NQM
Example 4, cont.
1. What do you need to know?
2. What do you already knowknow?
3. What can you showshow?
That all corresponding sides and angles are ≅≅≅≅
All sides are ≅≅≅≅ and m∠∠∠∠P = m∠∠∠∠N
∠∠∠∠PQR ≅≅≅≅ ∠∠∠∠NQM (vertical ∠∠∠∠s) and
∠∠∠∠R ≅≅≅≅ ∠∠∠∠M (3rd ∠∠∠∠s thm)
92°
92°Q
M
N
R
P
Example 4, cont.
Statement Reason
RP ≅≅≅≅ MN, RQ ≅≅≅≅ MQ, PQ ≅≅≅≅ NQ Given
m∠∠∠∠P = m∠∠∠∠N Given
If =, then ≅≅≅≅∠∠∠∠P ≅≅≅≅ ∠∠∠∠N
∠∠∠∠PQR ≅≅≅≅ ∠∠∠∠MQN If vertical ∠∠∠∠s, then ≅≅≅≅
∠∠∠∠R ≅≅≅≅ ∠∠∠∠M If 2∠∠∠∠s ≅≅≅≅, then 3rd ∠∠∠∠s ≅≅≅≅
∆∆∆∆PQR ≅≅≅≅ ∆∆∆∆NQM If corresp. ∠∠∠∠s and sides ≅≅≅≅, then figures ≅≅≅≅
92°
92°Q
M
N
R
P
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
A
B
C D
E
F J
K
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
Homework Check: 4.12: hypotenuse 4: base
6: acute isosceles 8: equiangular equilateral
10: B 12: A
14: F 16: acute isosceles
18: obtuse scalene 20: obtuse isosceles
22: sometimes 24: always
26: never
32: m∠∠∠∠1 = 50°, m∠∠∠∠2 = 40°, m∠∠∠∠3 = 45°
34: m∠∠∠∠A = 33°, m∠∠∠∠B = 66°, m∠∠∠∠C = 81°, acute
36: m∠∠∠∠W = 75°, m∠∠∠∠Y = 15°, m∠∠∠∠Z = 90°, right
38: x = 10 and m∠∠∠∠ = 109° 42: all three m∠∠∠∠ = 60°
48: see next
Homework Check: 4.1 (#47)
Homework Check: 4.1 (#48)