Transcript Aim: Are there any shortcuts to prove triangles are congruent?

Aim: Are there any shortcuts to prove
triangles are congruent?
Do Now:
In triangle ABC, the measure of angle
B is twice the measure of angle A and
an exterior angle at vertex C
measures 120o. Find the measure of
angle A.
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Congruence
Is ABCDE the exact same size and shape as STUVW?
S
A
T
B
E
D V
W
5 sides
5 angles
C
U
How would you prove that it is?
Measure to compare.
Measure what?
If the 5 sideAim:pairs
and 5 angle pairs
measure the
SAS – Triangle Congruence
Course: Applied Geometry
same, then the two polygons are exactly the same.
Corresponding Parts
Corresponding Parts – pairs of segments or angles
that are in similar positions in two or more polygons.
S
A
B
E
AB
BC
CD
DE
EA
IF
CORRESPONDING
PARTS
W
V
D
ARE
CONGUENT
T
THEN THE
C
U
A  S
 ST
POLYGONS ARE
 TU
B  T
CONGRUENT
 UV
C  U
 V
 VW Aim: SAS – Triangle Congruence
D Geometry
Course: Applied
 WS
E  W
Congruence Definitions & Postulates
Two polygons are congruent if and only if
1. corresponding angles are .
2. corresponding sides are .
Corresponding parts of congruent polygons
are congruent.
CPCPC
True for all polygons,
triangles our focus.
Corresponding Parts of Congruent Triangles
are Congruent.
CPCTC
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Model Problem
B
Hexagon ABCDEF
 hexagon STUVWX.
A
10 the value of the variables?
Find
B
A
AB and ST are
corresponding sides
x = 10
C
10
120
F
C
120
F
T
8
D
E
X
T
8
E
S
D
F & X are
corresponding ’s
x = 1200
S
U
X
X
U
2y
X
W
V
2y
ED and WV are corresponding
sides
V
2y = 8 y =W4
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Corresponding Parts.
Is ABC the exact same size and shape as GHI?
G
C
A
B
I
H
How would you prove that it is?
Measure corresponding sides and angles.
What are the corresponding sides? angles?
A  I
AC  GI
B  H
AB  IH
C  G
BC  GH
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Side-Angle-Side
SAS = SAS
I.
Two triangles are congruent if the two
sides of one triangle and the included
angle are equal in measure to the two
sides and the included angle of the other
triangle.
S represents a side of the triangle and
A represents an angle.
A’
A
B
C
B’
C’
If CA = C'A', A =A', BA = B'A',
then ABC = A'B'C'
If SAS  SAS ,
Aim: SAS – Triangle Congruence
Course: Applied Geometry
then
the triangles are congruent
Model Problem
Each pair of triangles has a pair of
congruent angles. What pairs of sides must
be congruent to satisfy the SAS postulate?
C
D
E
A
CE and EB; AE and ED
C
B
G
H
B
F
A
GH and BC ; FG and AB
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Model Problem
Each pair of triangles is congruent by SAS.
List the given congruent angles and sides for
each pair of triangles.
E
F
B
D
AB  DE; BC  EF , B  E
C
A
E
D
DE  DG; DF  DF ,
F
EDF  GDF
G
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Aim: Are there any shortcuts to prove
triangles are congruent?
Do Now:
Is the given information sufficient to prove
congruent triangles?
F
C
A
B
D
E
SAS = SAS
Two triangles are congruent if the two sides
of one triangle and the included angle are
equal in measure to the two sides and the
included
of Congruence
the other triangle.
Aim:angle
SAS – Triangle
Course: Applied Geometry
Side-Angle-Side
Is the given information sufficient to prove
congruent triangles?
F
C
A
A
B
F
B
C
D
E
E
D
B
D
C
D
A
B
A
Aim: SAS – Triangle Congruence
C
Course: Applied Geometry
Side-Angle-Side
Given that C is the
midpoint of AD and
AD bisects BE, prove
that ABC  CDA.
B
D
C
A
E
• C is the midpoint of AD means that
(S  S)
CA  CD.
• BCA  DCE because vertical
angles are congruent.
(A  A)
• AD bisects BE means that BE is cut in
(S  S)
to congruent segments resulting in
BC  CE.
The two triangles are congruent because
of SAS  SAS Course: Applied Geometry
Aim: SAS – Triangle Congruence
Side-Angle-Side
In ABC, AC  BC and CD bisects ACB.
Explain how ACD  BCD
C
A
D
B
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Side-Angle-Side
In ABC is isosceles. CD is a median.
Explain why ADC  BDC.
C
A
D
B
Aim: SAS – Triangle Congruence
Course: Applied Geometry
Sketch 12 – Shortcut #1
A
A’
B
B’
C
C’

Copied 2 sides and included angle:
AB  A’B’, BC  B’C’, B  B’
A’
B’
C’
Measurements
ABC
showed:
Shortcut for
proving
congruence in
Aim: SAS – Triangle Congruence
triangles:
 A’B’C’
SAS  SAS
Course: Applied Geometry
The Product Rule
Aim: SAS – Triangle Congruence
Course: Applied Geometry