1. 2.3: Exploring Congruent Triangles
GSE’s
M(G&M)–10–4 Applies the concepts of congruency by solving problems on or off a coordinate plane; or solves problems using congruency involving problems within mathematics or across disciplines or contexts.

2. Definitions
Congruent triangles: Triangles that are the same size and the same shape. A B C D E F
In the figure DEF  ABC
Congruence Statement: tells us the order in
which the sides and
angles are congruent

3. If 2 triangles are congruent:
The congruence statement tells us which parts of the 2 triangles are corresponding “match up”.
Means
3 Angles:
3 Sides:
ORDER IS VERY IMPORTANT

4. Meaning A  T, R  E, C  F AR  TE, RC  EF, AC  TF A R C T E
Example A R C T E F
In the figure TEF  ARC
Meaning
A  T, R  E, C  F
AR  TE, RC  EF, AC  TF

5. Congruent Triangles A Z B C X Y Write the Congruence Statement
Example 2
Congruent Triangles
Write the Congruence Statement A Z B C X Y
Example 3

6. Example 3 : Congruence Statement
Finish the following congruence statement:
ΔJKL  Δ_ _ _ N M L M J L K N

7. Definition of Congruent:
Triangles
(CPCTC)
Two triangles are congruent if and only if their corresponding parts are congruent. (tells us when Triangles are congruent)
Example:
Are the 2 Triangles Congruent. If so write
The congruence statement.

8. Ex. 2
Are these 2 triangles congruent? If so, write a congruence statement.

9. Reflexive Property Does the Triangle on the left have any
of the same sides or angles as the
triangle on the right?

10. SSS - Postulate
If all the sides of one triangle are congruent to all of the sides of a second triangle, then the triangles are congruent. (SSS)

11. Example #1 – SSS – Postulate
Use the SSS Postulate to show the two triangles are congruent. Find the length of each side.
AC = 5
BC = 7
AB =
MO = 5
NO = 7
MN =
By SSS

12. Definition – Included Angle
K is the angle between JK and KL. It is called the included angle of sides JK and KL.
What is the included angle for sides KL and JL? L

13. SAS - Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the triangles are congruent. (SAS) S A S S A S
by SAS

14. Definition – Included Side
JK is the side between J and K. It is called the included side of angles J and K.
What is the included side for angles K and L? KL

15. ASA - Postulate
If two angles and the included side of one triangle are congruent to two angles and the included side of a second triangle, then the triangles are congruent. (ASA)
by ASA

16. Identify the Congruent Triangles.
Identify the congruent triangles (if any). State the postulate by which the triangles are congruent.
by SSS
Note: is not SSS, SAS, or ASA.
by SAS

17. the 2 triangles are CONGRUENT!
AAS (Angle, Angle, Side)
If two angles and a non-included side of one triangle are congruent to two angles and the corresponding non-included side of another triangle, . . . F E D A C B
then
the 2 triangles are CONGRUENT!

18. the 2 triangles are CONGRUENT!
HL (Hypotenuse, Leg)
***** only used with right triangles****
If both hypotenuses and a pair of legs of two RIGHT triangles are congruent, . . . A C B F E D
then
the 2 triangles are CONGRUENT!

19. The Triangle Congruence Postulates &Theorems
AAS
ASA
SAS
SSS
FOR ALL TRIANGLES LA HA LL HL
FOR RIGHT TRIANGLES ONLY
Only this one is new

20. Summary Any Triangle may be proved congruent by: (SSS) (SAS) (ASA)
(AAS)
Right Triangles may also be proven congruent by HL ( Hypotenuse Leg)
Parts of triangles may be shown to be congruent by Congruent Parts of Congruent Triangles are Congruent (CPCTC).

21. Example 1 A C B D E F

22. Example 2
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? A C B F E D
No ! SSA doesn’t work

23. Example 3
Given the markings on the diagram, is the pair of triangles congruent by one of the congruency theorems in this lesson? D A C B
YES ! Use the reflexive side CB, and you have SSS

24. Name That Postulate
(when possible)
SAS
ASA
SSA
SSS

25. Name That Postulate
(when possible)
AAA
ASA
SSA
SAS

26. Name That Postulate SAS SAS SSA SAS Vertical Angles Reflexive Property
(when possible)
Vertical Angles
Reflexive Property
SAS
SAS
Vertical Angles
Reflexive Property
SSA
SAS

27. Let’s Practice B  D AC  FE A  F
Indicate the additional information needed to enable us to apply the specified congruence postulate.
For ASA:
B  D
For SAS:
AC  FE
A  F
For AAS:

28. Homework Assignment

29. Assignment