1. 5.5/5.6 Proving Triangles Congruent

2. SSS and SAS
What You'll Learn
You will learn to use the SSS and SAS tests for congruency.

3. 4) Construct a segment congruent to CB. 5) Label the intersection F.
SSS and SAS
4) Construct a segment congruent to CB.
5) Label the intersection F.
2) Construct a segment congruent to AC. Label the endpoints of the segment D and E.
1) Draw an acute scalene triangle on a piece of paper. Label its vertices
A, B, and C, on the interior of each angle.
6) Draw DF and EF.
3) Construct a segment congruent to AB. A C B D E F
This activity suggests the following postulate.

4. Triangles are congruent. sides three corresponding
SSS and SAS
Postulate 5-1
SSS
Postulate
If three _____ of one triangle are congruent to _____ _____________ sides of another triangle, then the two
Triangles are congruent.
sides
three
corresponding A B C R S T
If AC 
RT and
AB 
RS and
BC  ST
then ΔABC
 ΔRST

5. In two triangles, ZY  FE, XY  DE, and XZ  DF.
SSS and SAS
In two triangles, ZY  FE, XY  DE, and XZ  DF.
Write a congruence statement for the two triangles. X D Z Y F E
Sample Answer:
ΔZXY  ΔFDE

6. In a triangle, the angle formed by two given sides is called the
SSS and SAS
In a triangle, the angle formed by two given sides is called the
____________ of the sides.
included angle
C is the included
angle of CA and CB A B C
A is the included
angle of AB and AC
B is the included
angle of BA and BC
Using the SSS Postulate, you can show that two triangles are congruent if their corresponding sides are congruent.
You can also show their congruence
by using two sides and the ____________.
included angle

7. If ________ and the ____________ of one triangle are
SSS and SAS
Postulate 5-2
SAS
Postulate
If ________ and the ____________ of one triangle are
congruent to the corresponding sides and included angle of
another triangle, then the triangles are congruent.
two sides
included angle A B C R S T
If AC 
RT and
A 
R and
AB  RS
then ΔABC
 ΔRST

8. NO! On a piece of paper, write your response to the following:
SSS and SAS
On a piece of paper, write your response to the following:
Determine whether the triangles are congruent by SAS.
If so, write a statement of congruence and tell why they are congruent.
If not, explain your reasoning. P R Q F E D
NO!
D is not the included angle for DF and EF.

9. End of Section 5.5

10. ASA and AAS
What You'll Learn
You will learn to use the ASA and AAS tests for congruency.

11. It is the one side common to both angles.
ASA and AAS
The side of a triangle that falls between two given angles is called the ___________ of the angles.
included side
It is the one side common to both angles. A B C
AC is the included
side of A and C
CB is the included
side of C and B
AB is the included
side of A and B
You can show that two triangles are congruent by using _________ and the
___________ of the triangles.
two angles
included side

12. If _________ and the ___________ of one triangle are
ASA and AAS
Postulate 5-3
ASA
Postulate
If _________ and the ___________ of one triangle are
congruent to the corresponding angles and included side of
another triangle, then the triangles are congruent.
two angles
included side A B C R S T
If A 
R and
AC 
RT and
C  T
then ΔABC
 ΔRST

13. CA and CB are the nonincluded
ASA and AAS
CA and CB are the nonincluded
sides of A and B A B C
You can show that two triangles are congruent by using _________ and a
______________.
two angles
nonincluded side

14. If _________ and a ______________ of one triangle are
ASA and AAS
Theorem 5-4
AAS
Theorem
If _________ and a ______________ of one triangle are
congruent to the corresponding two angles and nonincluded side of another triangle, then the triangles are congruent.
two angles
nonincluded side A B C R S T
If A 
R and
C 
T and
CB  TS
then ΔABC
 ΔRST

15. ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to
ASA and AAS
ΔDEF and ΔLNM have one pair of sides and one pair of angles marked to
show congruence.
What other pair of angles must be marked so that the two triangles are
congruent by AAS?
If F and M are marked congruent, then FE and MN would be included
sides.
However, AAS requires the nonincluded sides.
Therefore, D and L must be marked. D F E L M N

16. End of Section 5.6