Parallel Lines and their Properties

Parallel Lines Definition
The definition for parallel lines states that if the lines lie in the same plane and if they don’t touch or meet the other lines at any point of the line, then these lines are termed to be parallel lines. If there are two parallel-lines PQ and RS, then it is said that the line PQ is parallel to the line RS.

When are two lines said to be parallel?
Two given lines are said to be parallel-lines if it satisfies any one of the following conditions:
• If they have a pair of alternate interior angles which are of equal measure.
• If they have a pair of corresponding angles which are of equal measure.
• If any one pair of interior angles which lie on the same side of the transversal are supplementary angles.

Constructing Parallel Lines
Now let us see how to construct parallel-lines. Draw a line AB and mark a point C at some place above the line AB.  Through C, draw a transverse line which cuts AB at D. The transverse line crosses C as well as AB. The line can cut the straight line at any angle.  With D as centre and with more than half of CD as radius draw an arc which cuts CD at E and AB at F.  With C as centre and with the same radius, draw a similar arc on the transverse line above the point C to cut the transverse line at G.

Now change the radius. The width of the lower arc that crosses the two lines AB and CD is taken as the compass width i.e. the radius.  With G as centre draw an arc on the upper arc to cut at H. Now a straight line is drawn through C and H.  We can see that the straight line CH is parallel to line AB.

Thus two parallel-lines AB and CH are constructed.

Properties of Parallel Lines
Consider there are two parallel-lines A and B which are cut by a transversal line. At the point of intersection of the transversal line with the two parallel-lines, the following properties will be met:
• The pair of acute angles formed in the parallel-lines is equal.
• The pair of obtuse angles formed in the parallel-lines is equal.
• The acute angles formed are supplementary to the obtuse angles formed.
• Equally measuring alternate interior angles are present.
• Equally measuring corresponding angles are present.
• The sum of the two interior angles which are present on the same side of the transversal is equal to 180 degrees.
• The sum of any acute angle with any obtuse angle is equal to 180 degrees.

Basic understanding of reflex angles

To define reflex angle, let us look at the following figures:

What is it that is different about these angles? Or in other words, what do we notice about these angles? Note that the measures of all the angles are greater than 180 degrees. Based on this understanding we now define reflex-angles as follows: An angle whose measure is more than 180 degrees and less than 360 degrees is called a reflex angle. Mathematically that can be written like this: If an angle measure α is such that, 180 < α < 360 degrees, then α would be a reflex-angle measure.

Now that we know what a reflex-angle is, the next most obvious question would be how can a reflex-angle be measured? Considering the fact that normally we use a set square or a protractor to measure angles, we know that the maximum angle that can be measured using a protractor or a set square is 180 degrees. So how can we measure reflex-angles?

How to measure a reflex angle?
Every normal angle, which is not a reflex-angle, has its corresponding reflex-angle.  Whether an angle is acute or obtuse, it would always have its corresponding reflex-angle. This can be seen in the examples below:
Example 1: 

Here the acute angle is 49 degrees and the corresponding reflex-angle is 311 degrees. The sum of these two angles is 49 + 311 = 360 degrees.
Example 2: 

Here we have an obtuse angle measuring 112 degrees and its corresponding reflex-angle measuring 248 degrees. The sum of these two angles is 112 + 248 = 360 degrees.

Therefore based on the definition of reflex angle, we can state that the sum of an angle and its corresponding reflex-angle is always 360 degrees. Thus if we want to measure a reflex-angle, we follow the following steps:
1. First we measure the corresponding acute or obtuse angle (say 112 degrees or 49 degrees in the above figures) = α degrees.
2. Now subtract the angle thus measured from 360 degrees. Thus our reflex-angle
= r = 360 - α degrees.

Where do we find reflex angles?

Reflex-angles are usually found in concave polygons.

A concave polygon would have at least one reflex-angle. Other examples of concave polygons are shown below:

Here we have a concave quadrilateral, a concave pentagon and a concave heptagon.

Geometry: Alternate interior angles

Definition: Alternate interior angles
Consider a pair of parallel lines is intercepted by a transversal. At each of the intersection points of the lines with the transversal, 4 angles are formed, making a total of 8 angles for the two lines. Each of these angles have a name or significance. Let us try to understand the following example of alternate interior angles.

The above figure shows two black lines intercepted by the red transversal. In the interior of the lines, four angles are formed namely, <5 a="a" also="also" alternate="alternate" and="and" angles.="angles." angles="angles" are="are" called="called" congruent.="congruent." green="green" if="if" interior="interior" is="is" of="of" other="other" p="p" pair="pair" similarly="similarly" the="the" these="these">
Theorem related to alternate interior angles:
When a pair of parallel lines is intercepted by a transversal, each pair of alternate interior angles thus formed are congruent. Therefore in the above figure, angle <5 and="and" angle="angle" congruent="congruent" is="is" p="p" to="to">
Examples of alternate interior angles:
The following pictures show examples of alternate interior angles:
Example 1:

In the above figure, the angles 76 and b are alternate interior angles. Therefore we can say that measure of angle b is 76. Similarly measure of angle a would be 104 since these two are also alternate interior angles and we know that alternate interior angles are congruent.

If one pair of alternate interior angles is acute, then the other pair of alternate interior angles has to be obtuse.  (Note an acute angle is an angle whose measure is less than 90 degrees and an obtuse angle is an angle whose measure is more than 90 degrees but less than 180 degrees)
Example 2:

In the above figure, the parallel lines are intercepted by a horizontal transversal. So here the purple dots are a pair of obtuse alternate interior angles and the pink dots are a pair of acute alternate interior angles. As we already know both the purple angles have to be congruent to each other and similarly both the pink angles also have to be congruent to each other.

Introduction to concept of median

What is median?
Median in math has two meanings. One is the geometric meaning and another is the statistical median.

Geometric median definition:
In a closed plane figure such as a triangle, the line segment that connects the midpoint of one side to the opposite vertex is called the median. See picture below:

The above picture shows a triangle ABC. D, E and F present on the sides AB, BC and CA, so that, AD = DB, CE = EB and AF = FC. The line segments AE, CD and BF are from the triangle ABC. The points where all the three geometric mid-segments intersect is called the centroid of the triangle also called the centre of gravity. In the above figure, O is the centroid of the triangle.

Statistical median definition:

In statistics it is a measure of central tendency. In a frequency distribution, the central value around which most of the values of the variable are centered is called the measure of central tendency. Of the various measures of central tendencies, the most popular are mean, middle number and mode. It is defined as middle value of the data set. For finding median we need to follow the following steps:

1. Arrange the data set in ascending or descending order.
2. If the number of entries is odd, then the middle value would be the (n+1/2)th value.
3. If the number of entries is even, then the middle value would be the average of the (n/2) th and the (n/2 + 1)th value.
4. The value found in step 2 or 3 is called the middle number value.

Sample problem:
1. Find the middle number of the following data set of marks obtained by 10 students in a class test of maximum 10 marks: 8, 8, 9, 5, 5, 6, 6, 7, 6, 4
Solution:
Step 1: Arrange the data in ascending order: 4, 5, 5, 6, 6, 6, 7, 8, 8, 9
Step 2: The number of entries is 10 which is an even number, so we move to step 3.
Step 3: The two middle values would be 10/2 = 5th and 10/2 + 1 = 5+1 = 6th value. Both these are 6. Therefore the middle number is 6.

Elementary math concepts

Elementary math covers all the basic operations, function and concepts in algebra. In elementary math is the main part covered is arithmetic operations. All the basic operations are presented in this area. This is covering all basic operation of addition multiplication, subtraction, and division. The elementary algebra covers all concepts from kindergarten level to middle school level. This is also help to solve for real life math problem.

Example: 4x+ 2 = 1

Elementary Math Concepts Covers:

Elementary math concepts contain this basic operations

Arithmetic Operations:

The real numbers have the following properties:

a + b= b +  a    ab  = ba                            (Commutative Law)

(a+ b)+ c= a+ (b + c)      (ab)c = a(bc)        (Associative Law)

a+(b +c)= ab +ac                                       (Distributive law)

Fractions:

To add two fractions numbers with the same denominator, we use the Distributive Law property:

a/b+c/b= 1/b*a+1/b*c  =1/b(a+c)  =a+c/b

To add two fraction with different denominators, we use a frequent denominator:

a/b+c/d =  ad+bc/bd

Factoring

Here we can make use of Distributive Law to increase certain algebraic conditions. In rare case we need to repeal this method (again using the Distributive Law) by factoring an expression as a product of simpler ones. The easiest condition occurs when the given expression has a common factor as given below...

3x(x-2)=3x2 – 6

Elementary Algebra Concepts Problems

Q 1 :   Find the larger number, -13 or -16?

Sol :  If a number has a negative sign, we dont conclude the number by ony seeing the value, here we have to consider the negative sign. If the value of a number with negative sign increases, the actual value of the number is decreses.

-16,-15,,-14,-13,-12,-111,-10,-9,-8,-7,-6,-5,-4,-3,-2,-1,0

so,-13 is larger than -16.

Q 2:  List all the integers between -2 and 4.

Sol :     -2,-1, 0, 1,2,3,4 these number are present in the -2 ,4

The -2, 4 between numbers are -1,0,1,2,3

Q 3:  Reduce 14/35.?

Sol :      14/35

We divide 7 on both sides

2/5

Q 4: Simplify 8 -: 2//3?

Sol :  We divide the given equation this is simple method the division inverse of multiplication

8

4*3   = 12

Q 5:  Simplify 1/2 + 2/3?

Sol  :      1/2+ 2/3

We take L.C. M   on 2, 3

1*3/2*3 +2*2/3*2

=3/6+4/6 =3+4/6

=7/6

Q 6.12x=4?

Sol :    x=4/12

x=1/3

Q 7:    3X+4y=5x-2y ?

Sol :         4Y+2y=5x-3x

6y=2x

y=2x/6

y=x/3

Q 8 :  x-4=8?

Sol : We add +4 on both sides

x=8+4

x=12

Q 9:  Factorize the given expression x2-9   ?\

Sol :  The general form of

(a2-b2)=(a-b)(a+b)

So, x2-9=  (x-3)(x+3)

Adjacent Angles

Definition for Adjacent Angles states that two angles are said to be adjacent, if:
• Both the angles are formed using the same side
• Both the angles have a same corner point i.e. the vertex
• Both the angles do not overlap on each other i.e. they should not have any interior point in common.

In simple words, Adjacent Angles Definition states that angles that are formed side by side using a common ray coming out of a common vertex in such a way that the common ray is between two other rays that forms the angles without any overlapping.

If two angles are given, they are said to be adjacent if they are only as per the Definition Adjacent Angles has stated.  There are certain scenarios when two angles do not satisfy the conditions stated in the Adjacent Angles Definition, they are:
1. Two angles share a common corner point or vertex but do not share a common side.
2. Two angles share the same side to form the angles, but do not have a common point at one of its corners.
3. Two angles given, in which one angle overlap the other.
The above cases that do not satisfy the conditions stated in the definition of adjacent angles can be declared as not adjacent. So these angles are not adjacent to each other.

Example of Adjacent Angles
An example of Adjacent-Angles helps to understand the concept in a better way. Consider three rays A, B, C coming out of the common vertex O. Two angles namely Angle x and Angle y are formed in such a way that angle x is formed between the sides OA and OB whereas angle y is formed using the sides OB and OC.  Here the vertex O is used as common and the side OB is used in common to form both the angles.

Adjacent Angles as Complementary Angles
When there are two adjacent-angles given with common vertex and common side, find the sum of the two angles.  If the total of the two angles is ninety (90) degrees and if it forms a right angle, then these adjacent-angles are said to be complementary and are termed as complementary angles. We can call it as adjacent complementary angles too.

Adjacent angles as Supplementary Angles
If two adjacent-angles are given, we can say that these adjacent-angles are supplementary angles if the total of the two adjacent-angles given is hundred and eighty (180) degrees and forms a straight angle. We can call it as adjacent supplementary angles.

Hypotenuse of a Right Triangle in brief

There are different types of triangle which we have learnt; one of them is the right angled triangle in other words a right triangle. A triangle in which one of the angles is a right angle that is 90 degrees is called a right triangle. The longest side of a right triangle is called the Hypotenuse of a Right Triangle and the other two sides are called the legs of the right triangle. We use the Pythagorean Theorem in finding the hypotenuse of a Right Triangle. The Pythagorean Theorem states that ‘the sum of the squares of the two sides (legs) of a right triangle is equal to the square of the hypotenuse’. Let us assume the lengths of the legs of a right triangle to be ‘a’ and ‘b’ units and the hypotenuse length to be ‘c’. By using Pythagorean Theorem, we can calculate the hypotenuse of a right triangle,
(Hypotenuse) ^2 = (sum of the squares of the sides (legs)^2
  c^2 = (a^2 +b^2)

This gives us the Hypotenuse of a Right Triangle formula, c = sqrt(a^2 + b^2)
Given the lengths of the sides or legs of a right triangle as 3 cm and 4 cm respectively, find the hypotenuse of the right triangle.  Here we are given the lengths of the two sides a = 3cm and b = 4cm, we need to find c. let us apply the Pythagorean Theorem, we get, c = sqrt(a^2+b^2). a^2= (3)^2 = 9 and b^2 = (4)^2 = 16, that gives us a^2+b^2 = 9 +16 = 25. So  c = sqrt(25) = +/- 5 , as length cannot be negative, the hypotenuse of the given right triangle is 5 cm.

Let us now learn how to calculate the Hypotenuse of a Right Triangle using the Pythagorean theorem given by hypotenuse = sqrt[(sum of the squares of the sides or legs)^2]. A ladder is placed against a wall of height 12 ft. The distance between the base of the ladder and the wall is 5ft, find the length of the ladder. In this problem, the triangle formed by the wall, the floor and the ladder is a right triangle and hence the length of the ladder would be the hypotenuse which we need to find. We are given the two lengths of the sides of the triangle which are 12ft and 5 ft respectively. We know c= sqrt(a^2 +b^2); here a = 12ft, b = 5 ft which gives us a^2 = 144 and b^2 = 25. So, c = sqrt(144 +25) = sqrt(169) = +/-13. The length of the ladder is 13 ft