Aim: SAS – Triangle Congruence Course: Applied Geometry
Do Now:
Aim: Are there any shortcuts to prove triangles are congruent?
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
Is ABCDE the exact same size and shape as STUVW?
Congruence
C
B
A
E D W
S
T
U
V
How would you prove that it is?
Measure to compare.
Measure what?
5 sides 5 angles
If the 5 side pairs and 5 angle pairs measure the same, then the two polygons are exactly the same.
Aim: SAS – Triangle Congruence Course: Applied Geometry
Corresponding Parts
CORRESPONDING PARTS
IF
AB BC CD DE EA
ST TU UV VW WS
THEN THE POLYGONS ARE
CONGRUENT
ARE CONGUENT
A B C D E
S T U V W
Corresponding Parts – pairs of segments or angles that are in similar positions in two or more polygons.
C
B
A
E D W
S
T
U
V
Aim: SAS – Triangle Congruence Course: Applied Geometry
Congruence Definitions & Postulates
Two polygons are congruent if and only if1. corresponding angles are .2. corresponding sides are .
Corresponding parts of congruent polygonsare congruent.
True for all polygons,triangles our focus.
CPCPC
CPCTC
Corresponding Parts of Congruent Trianglesare Congruent.
Aim: SAS – Triangle Congruence Course: Applied Geometry
Model Problem
Hexagon ABCDEF hexagon STUVWX. Find the value of the variables?
AB
C
DE
F
ST
U
VW
X
10
X
8
2y
120
AB and ST are corresponding sides
x = 10
x = 1200
F & X are corresponding ’s
ED and WV are corresponding sides2y = 8 y = 4
AB
C
DE
F
ST
U
VW
X
10
X
8
2y
120
Aim: SAS – Triangle Congruence Course: Applied Geometry
Corresponding Parts.
A B
C G
HI
Is ABC the exact same size and shape as GHI?
How would you prove that it is?
Measure corresponding sides and angles.
What are the corresponding sides? angles?
AC GIAB IHBC GH
A I B H C G
Aim: SAS – Triangle Congruence Course: Applied Geometry
Side-Angle-Side
I. 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 angle of the other triangle.S represents a side of the triangle and
A represents an angle.A
B B’C C’
A’
If CA = C'A', A =A', BA = B'A', then ABC = A'B'C'
If SAS SAS , then the triangles are congruent
Aim: SAS – Triangle Congruence Course: Applied Geometry
Model Problem
Each pair of triangles has a pair of congruent angles. What pairs of sides must be congruent to satisfy the SAS postulate?
E
D
A B
C
;CE and EB AE and ED
A
C
F
G
H
B
;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.
A
C
B
E
F
D
; ,AB DE BC EF B E
; ,DE DG DF DF
EDF GDF
G
D F
E
Aim: SAS – Triangle Congruence Course: Applied Geometry
Do Now:
Aim: Are there any shortcuts to prove triangles are congruent?
Is the given information sufficient to prove congruent triangles?
A B
C F
ED
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 angle of the other triangle.
Aim: SAS – Triangle Congruence Course: Applied Geometry
Side-Angle-Side
Is the given information sufficient to prove congruent triangles?
D E
FB
A C
A B
C F
ED
A
B
D
CA B
CD
Aim: SAS – Triangle Congruence Course: Applied Geometry
Side-Angle-Side
Given that C is the midpoint of AD and AD bisects BE, prove that ABC CDA. A
B D
E
C
• C is the midpoint of AD means that CA CD.
• BCA DCE because vertical angles are congruent.
• AD bisects BE means that BE is cut in to congruent segments resulting in BC CE.
The two triangles are congruent because of SAS SAS
(S S)
(A A)
(S S)
Aim: SAS – Triangle Congruence Course: Applied Geometry
Side-Angle-Side
In ABC, AC BC and CD bisects ACB. Explain how ACD BCD
A D B
C
Aim: SAS – Triangle Congruence Course: Applied Geometry
Side-Angle-Side
In ABC is isosceles. CD is a median. Explain why ADC BDC.
A D B
C
Aim: SAS – Triangle Congruence Course: Applied Geometry
Sketch 12 – Shortcut #1
SAS SASSAS SAS
B
A
C
Copied 2 sides and included angle:
AB A’B’, BC B’C’, B B’
Copied 2 sides and included angle:
AB A’B’, BC B’C’, B B’
B’
A’
C’
B’
A’
C’
Shortcut for proving congruence in triangles:
Measurements showed: ABC A’B’C’ABC A’B’C’
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