1) The document discusses the concept of congruence in geometry, specifically how congruence relates to triangles. It states that two triangles are congruent if specifying two sides and the angle between them (SAS), two angles and the side between them (ASA), or two angles and a corresponding adjacent side (AAS). 2) It also discusses properties of congruence, including that corresponding parts of congruent triangles (sides and angles) are congruent to each other. 3) The document provides examples justifying statements about angles and sides of triangles using properties of equality and congruence.

1. "Mathematics
teaches you
to think"

2. Properties of
Congruence

4. An example of congruence. The two
figures on the left are congruent, while the
third is similar to them. The last figure is
neither similar nor congruent to any of the
others. Note that congruences alter some
properties, such as location and
orientation, but leave others
unchanged, like distance and angles. The
unchanged properties are called
invariants.

5. Congruence
of triangles

8. Determining
Congruence

9. The shape of a triangle is
determined up to
congruence by specifying
two sides and the angle
between them (SAS),
two angles and the side between them
(ASA) or two angles and a
corresponding adjacent side (AAS).
Specifying two sides and an adjacent
angle (SSA), however, can yield two
distinct possible triangles.

10. Sufficient evidence for
congruence between two
triangles in Euclidean space
can be shown through the
following comparisons:

22. Angle-Angle-Angle
In Euclidean geometry, AAA (Angle-
Angle-Angle) (or just AA, since in
Euclidean geometry the angles of a
triangle add up to 180°) does not
provide information regarding the size
of the two triangles and hence proves
only similarity and not congruence in
Euclidean space.

23. However, in spherical
geometry and hyperbolic
geometry (where the sum
of the angles of a triangle
varies with size) AAA is
sufficient for congruence
on a given curvature of
surface.

24. If two triangles are
congruent, then each part of the
triangle (side or angle) is
congruent to the corresponding
part in the other triangle. This is
the true value of the concept;
once you have proved two
triangles are congruent, you can
find the angles or sides of one of
them from the other.

25. "Corresponding Parts
of Congruent Triangles are Congruent"
CPCTC is intended as an easy way
to remember that when you have
two triangles and you have
proved they are congruent, then
each part of one triangle (side, or
angle) is congruent to the
corresponding part in the other.

26. Justification
Using
Properties of
Equality and
Congruence

27. Properties Of
Equality For
Real Numbers

34. Properties Of
Congruence

38. EXAMPLES:

40. Statement Reason
1. 15y + 7 = 12 - 20y 1. Given
2. 35y + 7 = 12 2. Additive Property
3. 35y = 5 3. Subtractive Property
4. Y = 1/7 4. Division Property

42. Statement Reason Statement Reason
1. m ∠1 + m ∠2 =100 1. Given
2. m∠ 1 = 80 2. Given
3. 80 + m∠ 2 = 100 3. Substitution Property
4. m ∠2 = 20 4. Subtraction Property

44. Statement Reason Statement Reason
1. m∠ 1 + m∠ 3 = 80 1. Given
2. m∠ 1 = 40 2. Given
3. m∠ 3 = 40 3. Subtraction Property
4. m∠ 4 + m∠ 2 = 80 4. Given
5. m∠ 2 = 40 5. Given
6. m∠ 4 = 40 6. Subtraction Property
7. m∠ 3 = m∠ 4 7. Transitive Property

46. Statement Reason Statement Reason
1. m∠ 1 + m∠ 2 = 180 1. Given
2. m∠ 2 + m∠ 3 = 180 2. Given
3. m∠ 1 + m∠ 2 = m∠ 2 + m∠ 3 3. Transitive Property
4. m∠ 2 = m∠ 2 4. Reflexive Property
5. m∠ 1 = m∠ 3 5. Subtraction Property

47. Proofs are the heart of mathematics. If
you are a math major, then you must
come to terms with proofs--you must
be able to read, understand and
write them. What is the secret? What
magic do you need to know? The
short answer is: there is no secret, no
mystery, no magic. All that is needed
is some common sense and a basic
understanding of a few trusted and
easy to understand techniques.

48. PROOFS

55. Ending a proof
Sometimes, the abbreviation "Q.E.D." is written
to indicate the end of a proof. This abbreviation
stands for "Quod Erat Demonstrandum", which
is Latin for "that which was to be demonstrated".
A more common alternative is to use a square
or a rectangle, such as □ or ∎, known as a
"tombstone" or "halmos" after its eponym Paul
Halmos. Often, "which was to be shown" is
verbally stated when writing "QED", "□", or "∎" in
an oral presentation on a board.

56. EXAMPLE

64. THE END.