Blog Status - 26 January

Presently working on trigonometry chapters

Concept of Congruency of Triangles

Concept of Congruency
(From 9th Class Text)
(Used in Trigonometry Chapters of XI )

When two triangles have the same size, then they are said to be congruent triangles.

Two congruent triangles are equal in all respects and when one is placed on the other, both exactly coincide. This means each part of one triangle is equal to the corresponding part of the other.

If ABC and DEF are congruent triangles, when DEF is placed over ABC, both will coincide and this proves that they are congruent. This process of proof is known as proof by superposition.

But one need not check for all the six parameters (three sides and three angles) for proving congruency.

Conditions for Congruency

1. If two sides and the included angle of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the triangles are congruent.

2. If two angles and a side of a triangle are equal to the two angles and the corresponding side of another triangle, then the triangles are congruent.

3. If the three sides of the first triangle are equal to the corresponding three sides of the second triangle, then the triangles are congruent.

4. In case of right angled triangles, if the hypotenuse and a side of a triangle are equal to the corresponding side and hypotenuse of another triangle, then the triangles are congruent.

More briefly the rules are
(i) Two sides and the included angle (S.A.S.)
(ii) Two angles and corresponding side (A.A.S.)
(iii) Three sides (S.S.S.)
(iv) Right angle, hypotenuse, and one side (R.H.S.)

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Set: Explanation

Set is synonymous with the words, ‘collection’, aggregate’, ‘class’, and is comprised of elements.

The words ‘element’, ‘object’, and ‘member’ are synonymous.

Sets designated by specific letters.

N: natural numbers
Z : integers
Z⁺: positive integers
Q: rational numbers
Q⁺: positive rational numbers
R: real numbers
R⁺: positive real numbers
C: complex numbers

Description of a set

Sets can be described by roster method or set-builder method.

Roster method:
In this method, the set is described by listing all the elements within braces { }, separated by commas.

Example: {2,4,6,8,10}
It is a set having 5 elements.

Set-builder method:
In this method, a set is described by a property of x where x represents the elements. If the property of x is represented by P(x), the set description is given by
{x : P(x) is satisfied} or {x| P(x) is satisfied}

Example: {x| x is an even number less than or equal to 10}
This description will give {2,4,6,8,10} in roster form.

Types of sets

Empty set (ф)
A set is said t be empty or null or void set if it has no element and it is denoted by ф.

Singleton set
A set consisting of single element.

Finite set
A set is called a finite set if it is either void set or its elements can be listed (counted or labeled) by natural numbers 1,2,3 … and the counting of number of elements stops at a certain natural number of say (n).

The number of elements in a finite set (n) is called the cardinal number or order of a finite set A and is denoted by n(A).

Infinite set
A set who elements cannot be listed by the natural numbers however large the number may be is called an infinite set.

Equivalent set
Two finite sets are equivalent if their cardinal numbers or number of elements are same.

Equal set
Two sets A and B are equal if every element in A is a member of B and every element of B is a member of A.

Subset
When A and B are two sets, if every element of A is an element of B, then A is called a subset of B.

Universal set (U)
In discussions of sets, the superset that contains all other sets in discussion is called the universal set.

Power set
When A is a set, the collection or family of all subsets of A is called the power set of A and is denoted by P(A).

Power set is a set of subsets or elements of a power set are subsets of a set.
P(A) = {S: S is a subset of A}

If A is a finite set having n elements, the P(A) has 2ⁿ elements.


Complement of a set
If U is a universal set, the complement of a set A with respect to U is denoted as A’ or A^c or U – A . It is a set of those elements of U which are not in A.

A’ = {x| x є U, and x is does not belong to A}

Theorems on subsets

1. Every set is a subset of itself.
2. The empty set is a subset of every set.
3. The total number of subsets of a finite set containing n elements is 2ⁿ