Download Ways to Prove Triangles Congruent (ASA, SAS and SSS) SM1

Ways to Prove Triangles Congruent (ASA, SAS and SSS)
SM1—Section 9.2
Date ____________
Objectives:
1. Use the SSS Postulate, the SAS Postulate and the ASA Postulate to prove two triangles
congruent.
2. Identify corresponding parts, included angle and included side.
3. Understand why AAA and SSA are not sufficient to prove triangle congruence.
If you know that two triangles are congruent, you can conclude that the six parts of one triangle
are equal to the six parts of the other triangle. If you are not sure whether two triangles are
congruent, however, it is not necessary to compare all six pairs of parts. You need to check only
three pairs of parts.
Activity – Compare Triangles with Different Sides
Cut straws to makes side lengths of 4 cm, 5 cm and 6 cm. Make a triangle with the three lengths.
Trace the result here:
Mix the straws up, and make another triangle. If possible, make it different from the first. Trace
your new triangle onto patty paper. Compare this triangle to the first triangle you drew above
and write what you notice.
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In this activity, you can see that there is only one way to form a triangle given three side lengths.
In general, any two triangles with the same three side lengths must be congruent.
Side-Side-Side (SSS) Congruence Postulate
If three sides of one triangle are congruent to three sides of a second triangle, then the two
triangles are congruent.
L
A
N
M
C
B
If AB  LM and AC  LN and BC  MN , then
Postulate (SSS).
ABC 
LMN by the Side Side Side
O
M
N
P
Only two pairs of sides are marked as equal in MOP and NOP . Since OP  OP by the
Reflexive Property, we can conclude that MOP  NOP by the SSS Postulate.
Included Sides and Included Angles
EG is opposite F
E
EG is included between  E and G
FG is opposite  E
FG is included between F and G
F
G
Mark the included side:
I
1.
2.
3.
G
I
H
Z
F
J
P
E
Mark the included angle:
4.
5.
M
R
6.
Y
Z
T
X
N
O
S
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 triangles are congruent.
S
In this figure, you are given that ST  TP and
T
STR  PTR . You can also see that TR  TR
by the Reflexive Property.
R
Therefore,
RST 
RPT by SAS.
P
Name the congruent triangles and give a reason:
M
7.
L
A
D
8.
9.
Y
N
B
G
O
C
J
M
K
L
A
K
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Activity -- Does ASA work to show triangle congruence?
Suppose you cut two angles out of a piece of paper and place them at a fixed distance apart. Can
you form more than one triangle with a two given angles and given length as shown below?
50
30
5 cm
In the space below, using a protractor, draw a 30° angle. Measure and mark one side of that
angle to be 5 cm long. At that endpoint, construct a 50 ° angle.
Could you draw more than one triangle using this exact combination? Explain.
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From this activity, you can see that there is only one way to form a triangle given two angles and
the included side.
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 triangles are congruent.
A
R
B
C
S
Q
QS is included between Q and S . BC is included between B and C . QS  BC ,
B  Q and C  S . Therefore, QSR  BCA by Angle-Side-Angle (ASA).
Does AAA work to show triangle congruence?
W
S
B
X
V
T
R
A
F
C
As you can see in the first set of triangles, VWX and RST , all the corresponding angles are
congruent, but corresponding sides are not; therefore, the triangles are not congruent. In the
second picture, ABC and AGF again have corresponding angles congruent, but the sides are
not; therefore the triangles are not congruent. So, Angle-Angle-Angle DOES NOT prove
triangle congruence.
Name the congruent triangles and give a reason, if possible:
10.
T
Q
11.
S
12.
U
R
P
G
A
D
B
X
C
H
I
Y
Z
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Activity -- Does SSA work to show congruence of triangles?
Draw a horizontal line using a straightedge like the first drawing below. Connect two straws of
unequal length with at the ends, then position the straws so that they form a triangle with the
segment you just drew. Make the shorter straw on the right side. Anchor the longer straw with
tape, and label the vertices of the triangle A, B, and C, like the first picture.
Now swing the straw representing BC so that the end of the straw touches the segment again.
Label this point D.
B
A
B
C
A
D
C
Is triangle ABC congruent to triangle ABD?
Is SSA a valid test for determining triangle congruence? Explain your reasoning.
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In this activity, you can see that when the angle is not included between the two corresponding
congruent sides, two distinctly different triangles can be formed; therefore, you cannot use SSA
to prove two triangles congruent.
State the reason the following triangles can be proven congruent, if possible:
13.
14.
15.
16.