1. Congruent
Triangles
Triangles are congruent when all
corresponding sides and interior angles are
congruent. The triangles will have the same
shape and size, but one may be a mirror
image of the other.
3. SSS postulate
SSS (side, side, side) postulate
If three sides of a triangle are
congruent to its three corresponding
sides of another triangle, then the two
triangles are congruent.
4. AB ≅ ED ,
BC ≅ EF and
CA ≅ FD
∆ABC ≅ ∆DEF
Look at these two triangles
5. SAS postulate
SAS Postulate (Side-Angle-Side)
If two sides and the included
angle of one triangle are congruent
to the corresponding parts of another
triangle, then the triangles are
congruent.
6. Look at these triangles.
AC ≅ XZ
C ≅ Z
CB ≅ ZY
∆ABC ≅ ∆XYZ
7. EXAMPLE 1
Write a proof.
GIVEN
PROVE
STATEMENTS REASONS
BC ≅ DA, BC AD
∆ABC ≅ ∆ CDA
1. Given1. BC ≅ DAS
Given2.2. BC AD
3. BCA ≅ DAC 3. Alternate Interior
Angles Theorem
A
4. 4.AC ≅ CA Reflexive propertyS
9. Given: RS RQ and ST QT
Prove: Δ QRT Δ SRT.
Q
R
S
T
EXAMPLE 2
10. STATEMENT REASON ________
1. RS RQ; ST QT 1. Given
2. RT RT 2. Reflexive
3. Δ QRT Δ SRT 3. SSS Postulate
RQ S
T
EXAMPLE 2
11. ASA Postulate
ASA Postulate (Angle-Side-Angle)
If two angles and the included side
of one triangle are congruent to the
corresponding parts of another triangle,
then the triangles are congruent.
12. Look at these triangles.
B ≅ E
BC ≅ EF
C ≅ F
∆ABC ≅ ∆DEF
13. AAS Theorem
AAS (Angle-Angle-Side) Theorem
If two angles and a non-included
side of one triangle are congruent to two
angles and the corresponding non-
included side of a second triangle, then
the triangles are congruent.
14. Look at these triangles.
B ≅ E
C ≅ F
AC ≅ DF
∆ABC ≅ ∆DEF
18. EXAMPLE 5
STATEMENTS REASONS
1. EFG JHG 1. Given
2. EF HJ 2. Given
3. EGF JGH 3. Vertical angles
theorem
4. ∆EFG ∆JHG 4. AAS Theorem
19. Given: YR MA and AR RM
Prove: Δ MYR Δ AYR
Y A
R
M
Try to solve this.
20. CPCTC Theorem
•CPCTC states that if two
or more triangles are
proven congruent by any
method, then all of their
corresponding angles
and sides are congruent
as well.
21. Given: YR MA and AR RM
Prove: AY MY
Y A
R
M
Try to solve this.
22. To prove that triangles are
congruent we are going to use these
theorems and postulates.
1.The (SSS) Side-Side-Side postulate
2.The (SAS) Side-Angle-Side postulate
3.The (ASA) Angle-Side-Angle
postulate
4.The (AAS) Angle-Angle-Side
theorem
23. 2. GIVEN; DE CE, EA EB
PROVE; ∆DAB ∆CBA
1. GIVEN; circle with center
H
AHB FHB
PROVE; A F
H
A FB
D
E
C
B
A
Prove the following. ( 20 pts. )
24. Assignment
1.What is the HL theorem?
2.What is the LL theorem?
• Reference; Plane Geometry for Secondary Schools