Download Congruence and Triangles

4.2 Congruence and
Triangles
Geometry
Ms. Reser
Standards/Objectives:
Standard 2: Students will learn and apply
geometric concepts
Objectives:
 Identify congruent figures and
corresponding parts
 Prove that two triangles are congruent
Assignment
 4.2 Worksheet A and B
 Quiz 4.2 on page 210 to review for quiz
next time we meet.
Identifying congruent figures
 Two geometric figures are congruent if
they have exactly the same size and
NOT CONGRUENT
shape.
CONGRUENT
Triangles
Corresponding angles
A ≅ P
B ≅ Q
C ≅ R
B
A
Corresponding Sides
AB ≅ PQ
BC ≅ QR
CA ≅ RP
Q
CP
R
If Δ ABC is  to Δ
XYZ, which angle is
 to C?
Z
Thm 4.3
rd
3 angles thm
 If 2 s of one Δ are  to 2
s of another Δ, then the
3rd s are also .
Ex: find x
)
22o
87o
(4x+15)o
Ex: continued
22+87+4x+15=180
4x+15=71
4x=56
x=14
Ex: ABCD is  to HGFE, find x
and y.
C
A
9 cm
B
E
D
F
(5y-12)°
91°
B
D
113°
G
4x – 3 cm
C
4x-3=9
5y-12=113
4x=12
5y=125
x=3
y=25
A
86°
H
Thm 4.4
Props. of  Δs
 Reflexive prop of Δ  Every Δ is  to itself
(ΔABC  ΔABC).
 Symmetric prop of Δ  If ΔABC  ΔPQR, then
ΔPQR  ΔABC.
 Transitive prop of Δ  - If
ΔABC  ΔPQR & ΔPQR 
ΔXYZ, then ΔABC 
ΔXYZ.
A
B
C
P
Q
R
X
Y
Z
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
Start by labeling the
triangles. ∆AMT can
be the triangle on the
left and ∆CDN can
be the one on the
right. Do this now.
∆AMT  ∆CDN
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
A
M
D
N
T
C
Here’s what it should
look like. Next begin
by labeling the
drawing with tick
marks to keep track
of sides that are
congruent. Write
them down. Hint:
use ∆AMT  ∆CDN
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
A
M
D
N
T
C
AM 
MT 
TA 
Hint: use
∆AMT  ∆CDN
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
A
M
D
N
T
C
What about the angle
measures?
Hint: use
∆AMT  ∆CDN. The
letters of the
congruency
statement line up.
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
A
D
N
T
M
A 
M 
T 
C
What about the angle
measures?
Hint: use
∆AMT  ∆CDN. The
letters of the
congruency
statement line up.
Copy the congruent triangles shown then label the
vertices of your triangle so that ∆AMT  ∆CDN. Identify all pairs
of congruent corresponding angles and corresponding sides.
A
D
N
T
M
A  C
M  D
T  N
C
What about the angle
measures?
Hint: use
∆AMT  ∆CDN. The
letters of the
congruency
statement line up.
Some useful information

1.
2.
3.
4.
Often in proving any figures congruent, but
especially triangles . . . the following theorems are
especially helpful:
Reflexive Property – Segment  Segments i.e. BD
 BD with shared sides of a triangle (the side in
the middle).
Third Angles Theorem – when you know the other
two angles are there and congruent.
Vertical Angles Theorem – the X in the triangles
that look like bowties.
Last step is usually “Definition of Congruence”
because this is where the congruency statement
is given . . . ∆ABC  ∆DEF
Given: seg RP  seg MN, seg PQ  seg
NQ , seg RQ  seg MQ, mP=92o and
mN is 92o.
Prove: ΔRQP  ΔMQN
N
R
92o
Q
92o
P
M
Statements
Reasons
1.
2. mP=mN
3. P  N
4. RQP  MQN
5. R  M
6. ΔRQP  Δ MQN
1. given
2. subst. prop =
3. def of  s
4. vert s thm
5. 3rd s thm
6. def of  Δs