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

1
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.
2
Congruence
Is ABCDE the exact same size and shape as STUVW?

A
B
E
D
5 sides 5 angles
C
How would you prove that it is?
Measure to compare.
Measure what?
If the 5 side pairs and 5 angle pairs measure the
same, then the two polygons are exactly the same.
3
Corresponding Parts
Corresponding Parts pairs of segments or angles
that are in similar positions in two or more
polygons.
A
IF
B
CORRESPONDING PARTS
E
D
ARE CONGUENT
THEN THE POLYGONS ARE CONGRUENT
C
?S ?T ?U ?V ?W
ST TU UV VW WS
4
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
5
Model Problem
Hexagon ABCDEF ? hexagon STUVWX. Find the value
of the variables?
AB and ST are corresponding sides
x 10
?F ?X are corresponding ?s
?x 1200
ED and WV are corresponding sides
2y 8
y 4
6
Corresponding Parts.
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?
7
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
A
B
B
C
C
If CA C'A', ?A ?A', BA B'A', then DABC
DA'B'C' If SAS ? SAS , then the triangles are
congruent
8
Model Problem
Each pair of triangles has a pair of congruent
angles. What pairs of sides must be congruent to
satisfy the SAS postulate?
9
Model Problem
Each pair of triangles is congruent by SAS. List
the given congruent angles and sides for each
pair of triangles.
10
Aim Are there any shortcuts to prove triangles
are congruent?
Do Now
Is the given information sufficient to prove
congruent triangles?
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.
11
Side-Angle-Side
Is the given information sufficient to prove
congruent triangles?
12
Side-Angle-Side
Given that C is the midpoint of AD and AD bisects
BE, prove that DABC ? DCDA.
B
D
C
A
E

• C is the midpoint of AD means that
  CA ? CD.

(S ? S)

• ?BCA ? ?DCE because vertical angles are
  congruent.

(A ? A)

• AD bisects BE means that BE is cut in to
  congruent segments resulting in BC ? CE.
  

(S ? S)
The two triangles are congruent because of SAS ?
SAS
13
Side-Angle-Side
In ?ABC, AC ? BC and CD bisects ?ACB. Explain how
?ACD ? ?BCD
C
A
B
D
14
Side-Angle-Side
In ?ABC is isosceles. CD is a median. Explain
why ?ADC ? ?BDC.
C
A
B
D
15
Sketch 12 Shortcut 1
A
B
C
Copied 2 sides and included angle AB ? AB, BC
? BC, ?B ? ?B
Measurements showed
Shortcut for proving congruence in triangles
SAS ? SAS
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The Product Rule