Congruent triangles have congruent
sides and congruent angles.
The parts of congruent triangles that
“match” are called corresponding parts.
A
B C F
D
E
AB DF
BC FE
AC DE
A D B F
C E
TO PROVE TRIANGLES ARE CONGRUENT YOU DO NOT NEED TO KNOW ALL SIX
DFEABC
Before we start…let’s get a few things straight
A B
C
X Z
Y
INCLUDED ANGLEIt’s stuck in between!
Before we start…let’s get a few things straight
INCLUDED SIDEIt’s stuck in between!
A B
C
A B
C
Overlapping sides are congruent in each
triangle by the REFLEXIVE property
Vertical Angles are congruent
Alt Int Angles are congruent
given parallel lines
Side-Side-Side (SSS) Congruence Postulate
66
4 45 5
All Three sides in one triangle are congruent to all three sides in the
other triangle
Are these triangles congruent?
D
O
G
C
A
T
If so, write the congruence statement.
Side-Angle-Side (SAS) Congruence Postulate
Two sides and the INCLUDED angle
Are these triangles congruent?
If so, write the congruence statement.
C
A
T
H
A
T
Angle-Side-Angle (ASA) Congruence Postulate
Two angles and the INCLUDED side
Are these triangles congruent?
If so, write the congruence statement
B
I
G
T
O
E
Angle-Angle-Side (AAS) Congruence Postulate
Two Angles and One Side that is NOT included
Are these triangles congruent?
If so, write a congruence statement.
T
O
P H
A
T
The following slides will have
pictures of triangles. You are to
determine if the triangles are
congruent. If they are congruent,
then you should write a
congruence statement and state
which postulate you used to
determine congruency.
Δ_____ Δ_____ by ______
Determine if whether the triangles are
congruent. If they are, write a congruency
statement explaining why they are congruent.
ΔJMK ΔLKM by SAS
J K
LM
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they are
congruent.
Ex 4
R
P
S Q
ΔPQS ΔPRS by SAS
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they are
congruent.
Ex 5
R
P
S
Q
ΔPQR ΔSTU by SSST
U
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they are
congruent.R
T
S
ΔRST ΔYZX by SSS
Ex 2
Determine if whether the triangles are
congruent. If they are, write a congruency
statement explaining why they are congruent.
ΔGIH ΔJIK by AAS
G
I
H J
K
Not congruent.Not enough Information to Tell
R
T
S
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they are
congruent.
Ex 3
Determine if whether the triangles are
congruent. If they are, write a congruency
statement explaining why they are congruent.
ΔABC ΔEDC by ASA
B A
C
ED
Determine if whether the triangles are congruent. If they
are, write a congruency statement explaining why they are
congruent.
Ex 6
N
M
R
Not congruent.Not enough Information to Tell
Q
P
Warm up
Are they congruent, if so, tell how.1.
AAS
2.
Not
congruent
3.
Not
congruent
2-Column Proofs
• Going by the facts: definitions, properties, postulates, and theorems
• Numbering the statements and reasons
• Using logical order
Statements Reasons
1.
2.
3..
.
.
1.
2.
3..
.
.
Given: seg WX seg. XY, seg VX
seg ZX,
Prove: Δ VXW Δ ZXY
1 2
W
V
X
Z
Y
Proof
Statements Reasons
1. seg WX seg. XY 1. given
seg. VX seg ZX
2. 1 2 2. Vertical Angles
Congr Theorem
3. Δ VXW Δ ZXY 3. SAS Congr Postul
Given: seg RS seg RQ and seg ST
seg QT
Prove: Δ QRT Δ SRT.
Q
R
S
T
Proof
Statements Reasons
1. Seg RS seg RQ 1. Given
seg ST seg QT
2. Seg RT seg RT 2. Reflexive
Property
3. Δ QRT Δ SRT 3. SSS Congr Postulate
Example• Given that B C, D F, M is the
midpoint of seg DF
• Prove Δ BDM Δ CFM
B
D M
C
F
Proof
Statements Reasons
1. B C, D F 1. Given
2. M is the midpoint of 2. Definition of Midpt
seg DF
3. Seg DM seg FM 3. Reflexive Property
4. Δ QRT Δ SRT 4. AAS Congr Theorem
TRIANGLE
PROPORTIONALITY
THEOREMUsing similarity to find the missing parts of a triangle
TRIANGLE PROPORTIONALITY THEOREM
If a line parallel to one side of a triangle
intersects the other two sides, then it
divides those sides proportionally.
𝑨𝑫
𝑫𝑩=𝑨𝑬
𝑬𝑪
EXAMPLE 1:
Find the missing side length.
𝟖
𝟐𝟎=𝟏𝟖
𝒙
EXAMPLE 2:
Find the missing side length.
𝒙
𝟏𝟓=
𝟒
𝟏𝟎
EXAMPLE 3:
Find the missing side length.
𝒙
𝟏𝟓=
𝟐
𝟏𝟎
ON YOUR OWN:
1) Find the missing side length.
𝒙
𝟗=
𝟗
𝟏𝟓
ON YOUR OWN:
2) Find the missing side length.
𝟏
𝟒=𝟏
𝒙
ON YOUR OWN:
3) Find the missing side length.
𝟖
𝒙=𝟏𝟒
𝟑𝟓
ON YOUR OWN:
4) Find the missing side length.
𝒚
𝟏𝟐=𝟏𝟓
𝟏𝟎
Properties of
Parallelograms
•Both pairs of opposite
sides are parallel
Opposite sides are congruent
Opposite angles are congruent
Consecutive angles are supplementary
ABCD is a parallelogram. Find the lengths
and the angle measures.
1. AD
2. mADC
3. mBCD
B 8 C
A D
6545
E
5
8
110
70
4. Find the value of each variable
in the parallelogram.
4y
x = 5
2y4
2x – 6
y = 30
5. Find the measure of ∠D in the
parallelogram.
∠D = 115°
2x – 1x + 7
2x – 1x + 7
A
B
C
D
How to Prove
Quadrilaterals are
Parallelograms
How do you know if you have one?
1.BOTH pairs of opposite sides are parallel
2.BOTH pairs of opposite sides are congruent
3. BOTH pairs of opposite angles are congruent
4.ONE angle is supplementary to BOTH consecutive angles
5.diagonals BISECT each other
6. ONE pair of opposite sides are CONGRUENT & PARALLEL
60 120
120
6 6
60
50°
130°