"Mathematics teaches you to think" 

Properties of Congruence

In geometry, two figures are

congruent if they have the

same shape and size

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.

Congruence of triangles

Two triangles are congruent if their

corresponding sides are equal in length and their corresponding

angles are equal in size.

If triangle ABC is congruent to triangle DEF, the relationship can be written mathematically as:

In many cases it is sufficient to establish the equality of three corresponding parts and use one of the following results to deduce the congruence of the two triangles.

Determining Congruence

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.

The shape of a triangle is determined up to congruence by specifying two sides and the angle between them (SAS),

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

SAS (Side-Angle-Side):

If two pairs of sides of two triangles are equal

in length, and the included angles are

equal in measurement, then the triangles are

congruent.

SSS (Side-Side-Side):

If three pairs of sides of two triangles are equal in length, then

the triangles are congruent.

ASA (ANGLE-SIDE-ANGLE):

IF TWO PAIRS OF ANGLES OF TWO

TRIANGLES ARE EQUAL IN MEASUREMENT, AND THE INCLUDED SIDES

ARE EQUAL IN LENGTH, THEN THE TRIANGLES

ARE CONGRUENT.

THE ASA POSTULATE WAS CONTRIBUTED BY THALES OF MILETUS (GREEK). IN MOST SYSTEMS OF AXIOMS, THE THREE CRITERIA—SAS, SSS

AND ASA—ARE ESTABLISHED AS THEOREMS. IN THE SCHOOL MATHEMATICS

STUDY GROUP SYSTEM SAS IS TAKEN AS ONE (#15) OF

22 POSTULATES.

AAS (Angle-Angle-Side):

If two pairs of angles of two triangles are equal in

measurement, and a pair of corresponding non-included

sides are equal in length, then the triangles are congruent. (In British usage, ASA and AAS are usually combined into a single

condition AAcorrS - any two angles and a corresponding

side.)

RHS (RIGHT-ANGLE-HYPOTENUSE-SIDE):

IF TWO RIGHT-ANGLED TRIANGLES HAVE THEIR HYPOTENUSES EQUAL IN LENGTH, AND A PAIR OF

SHORTER SIDES ARE EQUAL IN LENGTH, THEN THE

TRIANGLES ARE CONGRUENT.

The SSA condition (Side-Side-Angle) which specifies two sides and a non-included angle (also

known as ASS, or Angle-Side-Side) does not by itself prove

congruence. In order to show congruence, additional information is required such as the measure of the corresponding angles and in

some cases the lengths of the two pairs of corresponding sides.

There are a few possible cases:

If two triangles satisfy the SSA condition and the length of the side opposite the angle is greater than or equal to the

length of the adjacent side, then the two triangles are

congruent. The opposite side is sometimes longer when the

corresponding angles are acute, but it is always longer

when the corresponding angles are right or obtuse.

Where the angle is a right angle, also known as the

Hypotenuse-Leg (HL) postulate or the Right-angle-Hypotenuse-Side

(RHS) condition, the third side can be calculated using the Pythagoras'

Theorem thus allowing the SSS postulate to be

applied.

If two triangles satisfy the SSA condition and the corresponding angles are acute and the length of the side opposite the angle is equal to the

length of the adjacent side multiplied by the sine of the angle, then the two triangles are

congruent.

If two triangles satisfy the SSA condition and the corresponding

angles are acute and the length of the side opposite the angle is greater than the length of the

adjacent side multiplied by the sine of the angle (but less than the

length of the adjacent side), then the two triangles cannot be shown

to be congruent. This is the ambiguous case and two different triangles can be formed from the

given information, but further information distinguishing them can

lead to a proof of congruence.

Angle-Angle-AngleIn 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.

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.

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.

"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.

Justification Using Properties of Equality and Congruence

Properties Of Equality For Real Numbers

Reflexive Property

For any number a, a = a.

Symmetric Property

For any numbers a and b, if a = b then b

= a.

Transitive Property

For any numbers a, b and c, if a = b and b = c, then a =

c.

ADDITION AND SUBTRACTION PROPERTIES

FOR ANY NUMBERS A, B AND C, IF A = B, THEN A

+ C = B + CAND A – C = B – C.

Multiplication and Division Properties

For any numbers a, b

and c, if a = b, then a c = b c and

if c 0, then a c = b c.

Substitution Property

For any numbers a and b, if a = b, then a may

be replaced with b in any equation.

Properties Of Congruence

REFLEXIVE PROPERTY OF CONGRUENCE

AB ≅AB

Symmetric Property of Congruence

If AB ≅CD , then CD ≅AB

Transitive Property of Congruence

If AB ≅CD and CD ≅EF , then AB ≅

EF

EXAMPLES:

Given: 15y + 7 = 12 - 20y

Conclusion: Y = 1/7

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

Given: m 1 m 2 ∠ + ∠

=100 m 1 ∠ = 80

Conclusion: m 2 ∠ =20

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

Given:

m∠ 1 = 40 m∠ 2 = 40

m∠ 1 + m∠ 3 = 80 m∠ 4 + m∠ 2 = 80

Conclusion: m∠ 3 = m∠ 4

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

Given: m∠ 1 + m∠ 2 = 180 m∠ 2 + m∠ 3 = 180

Conclusion: m∠ 1 = m∠ 3

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

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.

PROOFS

A proof is a demonstration that if

some fundamental statements (axioms) are assumed to be

true, then some mathematical statement is

necessarily true

PROOFS ARE OBTAINED FROM DEDUCTIVE REASONING,

RATHER THAN FROM INDUCTIVE OR EMPIRICAL ARGUMENTS;  A PROOF MUST DEMONSTRATE

THAT A STATEMENT IS ALWAYS TRUE (OCCASIONALLY BY

LISTING ALL POSSIBLE CASES AND SHOWING THAT IT HOLDS

IN EACH), RATHER THAN ENUMERATE MANY

CONFIRMATORY CASES.

An unproven proposition that is believed to be true is known as

a conjecture.

A statement that is proved is often called

a theorem.

Once a theorem is proved, it can be used as the basis to prove

further statements.

A theorem may also be referred to

as a lemma, especially if it is

intended for use as a stepping stone in

the proof of another theorem.

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.

EXAMPLE

Show that the sum of the first hundred whole

number is 5050.

Find the CONGRUENT FIGURES.

THE END.

THANK YOU!