Download 3.1-3.2 tri cong notes

10/6/14
The News
Triangle Congruence
Congruence: We have already determined that two
polygons are congruent when all of their corresponding
parts are congruent.
Now… we’re ready to learn some triangle short-cuts.
Good News!!
We know that ABC ≅ DEF because:
∠A
≅
∠D
BC = EF
∠D
AC = DF
≅ ∠F
3 sides (SSS)
Not-so-Good
News
AB = DE
∠B ≅ ∠E
In order to show that we have 2 congruent
triangles, it is only necessary to show 3 sets of
corresponding parts of the triangles are
congruent.
Triangles are fussy about WHICH 3 sets of
corresponding parts must be congruent
when showing triangles to be congruent.
Let’s check out the possibilities… then we can see which
ones work.
2 sides and the
INCLUDED angle
(SAS)
2 sides and the NONINCLUDED angle
(SSA or ASS)
ASS - The Donkey Theorem
ASS or SSA is tricky - Let look at an example.
3 angles (AAA)
2 angles and the
INCLUDED side
(ASA)
2 angles and the NONINCLUDED side
(SAA or AAS)
For a given triangle, there are 2
possible triangles with a nonincluded angle and 2 sides congruent
to the original (excluding a right
triangle).
So… does ASS ever work?
Yes - this special case is for right triangles only - it is called
HL (Hypotenuse-Leg)
What Works - There are 5 combinations
(and only 5) that work.
SSS
If three sides of one triangle are congruent to three sides of another triangle,
the triangles are congruent.
SAS
If two sides and the included angle of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent.
ASA
If two angles and the included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent.
AAS
If two angles and the non-included side of one triangle are congruent to the
corresponding parts of another triangle, the triangles are congruent.
HL
If the hypotenuse and leg of one right triangle are congruent to the
corresponding parts of another right triangle, the right triangles are
congruent.
What Doesn’t Work
AAA works fine to show
that triangles are the same
SHAPE (similar), but does
NOT work to show
congruent!
You can draw 2 equilateral
triangles that are the same
shape but not the same size.
ASS is referred to as
the Donkey
Theorem.
It does NOT work!!!
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10/6/14
Examples
More Examples
Are the two triangles on the left congruent?
If so, why?
YES - ∆ABC ≅ ∆DEF
In the figure to the left, ∠ N ≅ ∠ S, ∠M ≅
∠ R, and P is the midpoint of NS. Are the
two triangles congruent? If so, why?
In the figure to the left, AB = BC, and
AD = DC. Are the two triangles
congruent? If so, why?
Yes - ∆ ABD ≅ ∆ CBD by SSS
In quadrilateral PRST, RS = PT Are
the two triangles congruent? If so,
why?
Cannot be determined. More
information needed.
Yes - ∆ MPN ≅ ∆ RPS by SAA
More Examples
In the figure to the left, ∠ M ≅ ∠ O, and
MO = QP. Are the two triangles
congruent? If so, why?
Cannot be determined. More
information needed.
In the figure to the left, ∠ 1 ≅ ∠ 2,
∠ A ≅ ∠ E, and DC bisects AE. Are
the two triangles congruent? If so,
why?
≅
Yes - ∆ ABC
∆ EDC by ASA
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