Download 4.2 Apply Congruence and Triangles 4.3 Prove

4.1 Apply Congruence and Triangles
4.2 Prove Triangles Congruent by SSS, SAS
1.
2.
3.
Objectives:
To define congruent triangles
To write a congruent statement
To prove triangles congruent by SSS, SAS
Congruent Polygons
Congruent Triangles (CPCTC)
Two triangles are congruent triangles if
and only if the corresponding parts of
those congruent triangles are congruent.
Congruence Statement
When naming two congruent triangles, order is very
important.
Example
Which polygon is congruent to ABCDE?
ABCDE  -?-
Properties of Congruent Triangles
Example
What is the relationship
between C and F?
D
75
F
C
30
E
A
75
30
B
Third Angle Theorem
If two angles of one triangle are congruent
to two angles of another triangle, then the
third angles are also congruent.
Congruent Triangles
Checking to see if 3 pairs of corresponding
sides are congruent and then to see if 3
pairs of corresponding angles are
congruent makes a total of SIX pairs of
things, which is a lot! Surely there’s a
shorter way!
Congruence Shortcuts?

Will one pair of congruent sides be sufficient? One
pair of angles?
Congruence Shortcuts?

Will two congruent parts be sufficient?
Congruent Shortcuts?
Will three congruent parts be
sufficient?
 And if so….what three parts?

Investigation…

… Using 2,3,4 Triangles
Side-Side-Side Congruence Postulate
SSS Congruence Postulate:
If the three sides of one triangle are congruent to the
three sides of another triangle, then the two triangles
are congruent.
SSS Congruence Postulate
Using a 2-Column Proof!
C
Is ABC is congruent to
ABD? Why or why
not?
A
B
D
Example
Decide whether the triangles are congruent. Explain
your reasoning.
Investigation…

…

Part 2
Congruence Shortcuts
Side-Angle-Side (SAS) Congruence Postulate:
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of
another triangle, then the two triangles are congruent.
Can we prove?...yet?
What else would we need?
Can we Prove Triangle Congruence?
Another Proof…?
Congruent Shortcuts?

Will three congruent parts be sufficient?
Congruent Shortcuts?

Will three congruent parts be sufficient?
Congruent Shortcuts?

Will three congruent parts be sufficient?
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Which case do we have? (SSS,SAS…)
(They may not all work though!!!!)
Investigation: Shortcuts
Shortcuts?:
√
SSS
AAS
√
SAS
AAA
ASA
ASS
Well, we know that SSS is a valid
shortcut, and I’ll give you the hint
that 2 others in the list do not
work.
To test the remaining 5, we will use
our protractor and ruler. If the
shortcut works, one and only one
triangle can be made with those
parts.
Congruence Shortcuts
Side-Side-Side (SSS) Congruence Postulate:
If the three sides of one triangle are congruent to the
three sides of another triangle, then the two triangles
are congruent.
Congruence Shortcuts
Side-Angle-Side (SAS) Congruence Postulate:
If two sides and the included angle of one triangle are
congruent to two sides and the included angle of
another triangle, then the two triangles are congruent.
Congruence Shortcuts
Angle-Side-Angle (ASA) Congruence Postulate:
If two angles and the included side of one triangle are
congruent to two angles and the included side of
another triangle, then the two triangles are congruent.
Congruence Shortcuts
Angle-Angle-Side (AAS) Congruence Theorem:
If two angles and a non-included side of one triangle
are congruent to the corresponding two angles and
the non-included side of another triangle, then the two
triangles are congruent.
Remember our justifications…
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





Vert angles 
Linear Pair are Supp.
Alt Int Angles congruent
Defn of midpt
Defn of angle bisector
Right angles are congruent
Defn of 
…..
Classwork
From 4.1
 3 - 12
 32 – 35
 38 – 40
 46 – 48*

From 4.2
1 - 4
 8 - 17
 22 - 30
 41 - 42

From 4.3
1 - 7
 18
 22 - 25
 29
