properties of congruence
TRANSCRIPT
"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!