This document provides information on proving triangle congruence using various postulates and properties. It discusses the six corresponding parts used to determine if two triangles are congruent, as well as five postulates for proving congruence: SSS, SAS, ASA, SAA/AAS, and the third angle theorem. Examples are given of applying each postulate, along with exercises to identify the postulate used and complete triangle congruence proofs. Key details include identifying the six corresponding parts of triangles as sides and angles, discussing the five postulates for proving congruence based on sides and angles, and providing examples of setting up triangle congruence proofs.

2. ~3.1 Congruent Triangles~
•Basis used to determine congruence in triangles
•Triangles have six parts by which we determine whether two
triangles are congruent:
•The three sides of the triangle
•The three angles of the triangle
~The two triangles are congruent if the corresponding parts are
equal.

3. • The corresponding parts of two triangle are the parts
which “match” or the parts which hold the same
relative or ”corresponding” positions with respect to
the other parts of the triangle.
The sides and angles that match are called
corresponding sides and corresponding angles.

4. A
B C
X
Y Z
• For both triangles, segment AB and XY are the shortest
sides, segment BC and YZ are the “medium” sides and
segment AC and XZ are the longest sides.
• Corresponding sides: segment AB & XY, BC & YZ
and AZ corresponds to XZ.

5. • We can say that:
∆ABC = ∆XYZ ∆CAB = ∆ZXY
∆BCA = ∆YZX ∆BAC = ∆YZX

6. • Example 01
~ ∆YES and ∆ART are congruent.Identify the
corresponding parts.
Y
E S
A
R
T
~CORRESPONDING PARTS ARE:
segment YE = segment RT angle Y = angle R
segment YS = segment RA angle S = angle A

7. ~Properties of equality and equivalence
relations~
1.Addition property - - - - if a = b,then a + c = b + c
or, if a = b & c = d, then
a + c = b + d
2. Subtraction property- - -if a = b, then a – c = b – c
or, if a = b & c = d, then
a – c = b – d
3.Multiplication property-if a = b, then ac = bc
if½a=½b, then a = b

8. 4.division property - - - - if a = b, then a/c = b/c
if a = b, then ½a = ½b
(halves of equals are equal).
• For the next properties, we will use congruence as
equivalence relations.
5.Reflexive property- - - -any number or figure is
congruent to itself.
Ex. Segment AB = segment AB.
6.Symmetric property- - - -members on both sides of the
congruence symbol are
interchangeable.

9. 7.Transitive property- - - - figures that are congruent to
the same figure are also
congruent to each other.

10. Exercises:
Say if what property is applied in the following.
1.If X = Y and N=M, then X-N=Y-M
2.AB = CD , then AB= CD
2 2
3. ST = ST
4.IT = TO, then TO = IT
5.GH=LM and LM= OP, then GH=OP

11. Answers:
1.Subtraction Property
2.Multiplication Property
3. Reflexive Property
4.Symmetric Property
5.Transitive Property

12. ~Rules used to prove Congruent Triangles~
1. Using all six corresponding parts
Postulate 25 CPCTC postulate
-if all six pairs (angles and sides) of two triangles
are congruent , the two triangles are congruent.
CPCTC stands for corresponding parts of congruent
triangles are congruent.

13. M
N O
P
Q R
The corresponding parts of ∆MNO and ∆PQR are
congruent. There may be times, however , when not all
the measures of the sides and the angles are given.
MN=PQ
NO=QR
MO=PR
angle M = angle P
angle N = angle Q
angle O = angle R

14. 2.Three sides of the triangles
Postulate 26 side-side-side (SSS) postulate
~ if three sides of one triangle are congruent to three
sides of another triangle, the two triangles are
congruent.

15. 4 4
8 8
5 5
A
B C
D
E F
Thus, by the SSS Postulate

16. 3.Two sides and an included angle
• Postulate 27 side-angle-side (SAS) postulate
~ if two sides and the included angle of one triangle are
congruent to two sides and the included angle of
another triangle, the two triangles are congruent.
• An included angle created by two sides of a triangle.

17. G
H I
K
L M
55
Example:
GH=KL
HI= LM
Thus , ∆GHI = ∆KLM by the SAS postulate.

18. 4.Two angles and the included side
• Postulate 27 angle-side-angle (ASA) postulate
~if two angles and the included side of one triangle are
congruent to two angles and the included side of
another triangle, the two triangles are congruent.
An included side is a side that is between two angles.
for example, in the figure used of another triangle, the
two triangles are congruent.

19. O
N
Q
P
Thus, by ASA Potulate

20. 5.Two angles and non-incuded side
Postulate 28 side-angle-angle (SAA) or angle-angle-side
(AAS) postulate
~if two angles and one non-included side of one triangle
are congruent to two angles and the corresponding
non-included side of another triangle, the two
triangles are congruent.

21. R
S T
U
W V
Thus, by the AAS or SAA Postulate

22. 6.Third angle postulate
Postulate 29
~ if two angles of one triangle are congruent to two
angles of a second triangle , then third angles of the
triangles are congruent.

23. A
B C
X
Y Z
Since m A=m X and m B=m Y,
Then m C= m Z
Therefore by TA Postulate ∆ABC=∆XYZ

24. N.b.
There are also postulate that do not work in
proving that triangles are congruent.
1. Angle-Angle-Angle(AAA)
AAA can show that the triangles are of the same
shape( similar) but it does not necessarily mean
that the triangles are congruent.
2. Angle-Side-Side (ASS)
This does not show congruency.

25. Summary:
6 Postulates to prove that two triangles are congruent.
a. CPCTC Postulate – Corresponding angles and side of two
triangles are congruent.
b. SSS Postulate - All corresponding sides of two triangles
are congruent.
c. SAS Postulate – two sides and an included angle of two
triangles are congruent.
d. ASA Postulate - two angles and an included side of two
triangles are congruent.
e. SAA/AAS Postulate - two angles and a non- included
side of two triangles are congruent.
f. TA Postulate - if two corresponding side of two triangles
are congruent, then the third angles of the triangles are
congruent.

26. Exercise:
A.Give the name of the Postulate that is ask in each
of the ff:
1. Segment AB= segment DE; segment BC = segment EF;
angle B= angle E
2. Angle R= angle H; segment RS= segment HI ; angle I=
angle S
3. Segment JK= segment MN ; segment KL= segment
NO; segment JL= segment MO

27. B. Write the corresponding parts needed:
X
Y
Z
O
1. ∆YXO=∆_____
2. YO= _____
3. angle X= _____
4. angle O= _____
5. YX= _______
6.∆YOZ=∆_____

28. Answers:
A.
1. SAS Postulate
2. ASA Postulate
3. SSS Postulate
B.
1. ∆ YZO
2. YO
3. angle Z
4. angle O
5. YZ
6. ∆ YOX

29. 3.2 Proving triangle congruence
• How to make a proof ?
1.Make a diagram of the figure and mark the given
information about the sides and angles and othe
relevant parts.
2.Select which rule (in this case SSS, SAS, AAS/SAA)
You will be using in proving the statement .
3.In one column, write the statements and in another
column write the reasons for the statements.

30. T U V
W X
Given: angle TUW = angle VUX
angle T = angle V
U is the midpoint if segment TV
Prove: ∆TUW = ∆VUX

31. 1.Angle TUW = VUX (angle)
2.Angle T = angle V (angle)
3.U is the midpoint of segment
TV.
4.Segment TU = UV
(included side)
5.∆TUW = ∆VUX
1.Given
2.Given
3.Given
4.Definition of a midpoint-it
evenly divides the segment into
two.
5.ASA postulate
STATEMENT REASONS

32. Exercises:
A. Complete each proof.
1. given: segment SU bisects angle TSV
segment SU is perpendicular to segment TV
PROVE:∆SUV = ∆SUT
T
S U
V

33. a. segment SU bisects angle TSV
b. Angle TSU = angle VSU
c. Segment SU =segment SU
d. Segment SU is perpendicular to
segment TV
e. Angle SUT & angle SUV are
right triangles
f. Angle SUT=90°
Angle SUV=90°
g. Measure of angle SUT = measure
of angle SUV
h. Angle SUT = angle SUV
i. ∆SUV = ∆SUT
a.
b.
c.
d.
e.
f.
g.
h.
i.
STATEMENTS REASONS

34. a. segment SU bisects angle TSV
b. Angle TSU = angle VSU
c. Segment SU =segment SU
d. Segment SU is perpendicular to
segment TV
e. Angle SUT & angle SUV are
right triangles
f. Angle SUT=90°
Angle SUV=90°
g. Measure of angle SUT = measure
of angle SUV
h. Angle SUT = angle SUV
i. ∆SUV = ∆SUT
a. Given
b. Definition of Angle bisector
c. RPC
d. Given
e. Definition of perpendicular
angles
f. Definition of right angles
g. Substitution
h. Definition of Congruent
Angles
i. ASA Postulate
STATEMENTS REASONS