How can we tell when two triangles are congruent?
We know that when two figures are congruent, one can be mapped onto the other and will fit perfectly. All of the corresponding sides of the two figures are congruent and all of the corresponding angles are congruent. In the case of congruent triangles, if we know that two triangles are congruent, then we can make six congruence statements about the parts of the triangles (the three corresponding sides are congruent and the three corresponding angles are congruent).
Write the six congruence statements about corresponding sides and angles given that

ABC
 
DEF.
E
B
A
D
C
F
If we know that all six corresponding parts of two triangles are congruent, then we know that the two triangles are congruent. Do we need ALL six pieces of information, or are there some shortcuts?

A B m AB = 2.00 in.
C D m CD = 3.00 in.
E F m EF = 4.00 in.
G H m GH = 5.00 in.
m IJ = 6.00 in.
I J
Suppose you wanted to make scalene triangles using only the segments above. Work with a small group to answer the questions below.
1) How many different triangles could you make? How did you decide?
2) List all the triangles you could make.
Using only one side of some blank paper (you may need more than one sheet), use your compass and straightedge to construct all the triangles you named in #2. Once you have constructed all your triangles, cut them out and label the lengths of the sides (as shown below)
4
9
10

3) Following your teachers’ directions, have one group member take each triangle to the area in your classroom labeled with the dimensions of each triangle. Compare the shape and size of your group’s triangle with the other groups’ triangles. Compare the measures of the angles of your triangles with other groups’ triangles. Record your findings below.
Fill in the blanks to complete a conjecture based on your findings:

So far in this investigation, you have learned that you do not have to know that all six corresponding parts are congruent in order to determine that two triangles are congruent.
If you only know that the three corresponding sides are congruent, the triangles turn out to be congruent as well. We call this shortcut SSS (side-side-side).
Do you think the following statement is true? Why or why not?
If the three angles of one triangle are congruent to the corresponding three angles of another triangle, then the triangles are congruent.
In your small groups, create a triangle whose angles measure 30

, 60

, and 90

. Compare your triangle to other groups’ triangles. What do you notice?
What if you knew some combination of angles and sides of one triangle were congruent to the corresponding angles and sides of another triangle? Let’s investigate various possibilities.
SAS—two sides and the included angle of one triangle are congruent to the corresponding two sides and the included angle of another triangle

A
ASA—two angles and the included side of one triangle are congruent to the corresponding two sides and included angle of another triangle
AAS—two angles and a non-included side of one triangle are congruent to the corresponding two angles and non-included side of another triangle
SSA—two sides and a non-included angle of one triangle are congruent to the corresponding two sides and non-included angle of another triangle
45

5 inches B
4 inches

Use your compass to complete

ABC by locating point C on the top ray of angle A so that BC = 4 inches (use the segment marked 4 inches). Cut out your triangle. Compare it with other people’s triangles. Are they all congruent? What does this tell you about one of our combinations?
Given MN below, draw a ray that starts at point M and goes through point P (above
MN ) to form

PMN so that m

PMN = 30

. Now choose a point Q (also above MN ) to form

QNM so that m

QNM = 45

. Call the point where MP and NQ intersect Point reasoning.
O.
M N
What does this information tell you about one of our combinations? Explain your

In your groups, determine a way to test whether SAS and AAS work. Completely describe how you determined if they work or not.
SAS
AAS
Suppose you have two right triangles and you know that their hypotenuses are congruent.
You also know that a leg of one of the triangles is congruent to the corresponding leg of the other triangle. What conclusion can you draw about the other legs of the two triangles? What does this say about the two triangles? Explain your reasoning.
Using your conclusions, complete the following statement: If two right triangles have congruent hypotenuses and a pair of congruent corresponding legs, then____________
______________________________________________________________________

You can conclude that two triangles are congruent if you have the following situations:
Shortcut abbreviation
Picture
SSS

1. In the figure below, mark one more pair of corresponding parts of the two triangles congruent that will ensure that the two triangles are congruent. State how you know the two triangles are congruent.
E
A
B
D
F
C
2. Suppose you knew that the hypotenuses of two right triangles were congruent. Are the two triangles necessarily congruent? Explain why they are or give an example to show that they are not.
3. In the figure below, can you conclude that ΔGHI 
ΔJKI? Explain why or why not.
G H
I
GH KJ
GI bisects HK
K
J

1. Mark 3 parts of ΔLMN congruent to 3 parts of ΔOPQ so that ΔLMN 
ΔOPQ. State the congruence postulate or theorem that you used to determine that the two triangles were congruent.
N
M
Q
L
P
O
2. Mark 3 parts of ΔLMN congruent to 3 parts of ΔOPQ so that ΔLMN is not necessarily congruent to ΔOPQ. Explain how you know that the two triangles are not necessarily congruent.
M
L
N
P
Q
O

3. Given parallelogram ABCD with AB DC and AD BC , prove that the opposite sides of the parallelogram are congruent (AB

CD; AD

CB).
D
A
C
B
4. Given the triangles below, can you conclude that ΔEFG

KLM? Why or why not?
E
F
J
H
I K
L
G
M

5. ABCD is a square. Prove that ΔABC

ΔADC two different ways.
B C
A
D
6. Draw and mark parts congruent in two triangles ΔPQR and ΔLMN so that ΔPQR

ΔLMN by AAS.