Congurence

The term congruent comes from the Latin word congruere, meaning "to come together." It therefore carries with it the idea of superposition, the idea that one of two congruent figures can be picked up and placed on top of the other with all parts coinciding. In the case of congruent triangles the parts would be the three sides and the three angles.

Two triangles are congruent if two sides and the included angle of one are congruent to two sides and the included angle of the other. This can be proven by super-imposing one triangle on the other. They have to match. Therefore the third side and the two other angles have to match. Modern authors typically make no attempt to prove it, taking it as a postulate instead.

Inequalities in Algebra

In general equality means that something should be equal ( = ). Inequality means that something should not be equal ( ≠ ). So It may be greater than ( > ) or less than ( < ). Those symbols are going to use in solving inequality problems.

Solving inequalities in algebra:

1.Addition principles for inequalities in algebra:

If a > b then a + x > b + x .

Example:

For solving inequalities in algebra, consider the following example.

Solve x + 5 > 10

Solution:

Using the addition principle, add -5 to both sides of the inequality.

x + 5 -5 > 10 – 5

So , x > 5. Answer.

2. Multiplication principle for inequalities.

If a > b and x is positive, then ax > bx . If a > b and x is negative, then ax <>

Example:

Solve: - 4x <>

Solution:

Using the multiplication principle, divide 4 both on sides, then reverse the signs.

( -4x ) / 4 <>

- x <>

So x > 2. Answer

These are the examples for solving the inequalities in algebra.

Symmetry

In formal terms, we say that a mathematical object is symmetric with respect to a given mathematical operation, if, when applied to the object, this operation preserves some property of the object. The set of operations that preserve a given property of the object form a group. Two objects are symmetric to each other with respect to a given group of operations if one is obtained from the other by some of the operations (and vice versa).

Types of Symmetry:

There are two types of Symmetry as follows:

• Symmetry in Geometry
• Symmetry in Mathematics

Symmetry in Geometry: Symmetry definition in geometry it means a sub-group. In isometrics consists of three or two dimensional space. In following operations

1. Reflectional Symmetry(FLIP)
2. Rotational Symmetry (TURN)
3. Translational Symmetry (SLIDE)

Symmetry in Mathematics: In mathematical operation, to apply the object into operation. The set of operations to form a group. Two object form a group of operations. To apply the objects into symmetry. So it is called a symmetry definition in mathematics.

Slope of Parallel Lines

Slope of a line is the value of the angle that a straight line makes with the positive direction of x-axis in the anticlockwise sense. Slope is also refer to as Gradient.

Basically for any straight line y = mx + b , m is referred to as slope.

Parallel lines are the lines that lie in the same plane have the property that they never intersect each other. They never meet at a point. They are always the same distance apart.

Slope of Parallel lines : Property

Parallel lines have the value of the angle made with the x axis always same.. they always have the same slope.

Slopes of two parallel lines are equal.

Let the equation of two lines be y=(m₁)x + c₁ and y=(m₂)x + c₂ where m₁ and m₂ are slopes of the lines. The two lines are parallel if and only if m₁ = m₂.

Quadratic Equation Formula

A quadratic functions in the variable of x is an equations of the general form ax² + b x + c = 0, Where "a","b","c" are real numbers, a not equal to Zero. 2x² + x – 300 = 0 is a quadratic equations.

The Simplifying Quadratic equations, the general form is ax² + bx + c, where "a", "b", and "c" are just numbers; they are known as the "numerical coefficients". The Formula is derived by the process of completing the square.

Simplifying Quadratic Equations formula:

The solutions of any quadratic equations, ax² + bx + c = 0 is given by the following formula, called the quadratic formula:

-b+(√b²-4ac) /2a

For simplifying Quadratic equations Formula to work, the equations must be arranged in the form "(quadratic) = 0". Also, the "2a" is the denominator Formula in underneath everything above, not just the square root. And it's a 2a under there, not just a plain 2.