Some useful things about Lines and Angles

        Some useful things about Lines and Angles

 

1. Two angles are said to be supplementary if the sum of their measures is 180 degree.

2. Two angles are said to be complementary if the sum of their measures is 90 degree.

3. Two angles with the same vertex, one arm common and other arms lying on opposite sides of the common arm, are called adjacent angles.

4. Two adjacent angles are said to form a linear pair of angles if their non-common arms are two opposite rays.

5. If a ray stands on a line then the sum of the adjacent angles so formed is 180 degree.

6. The sum of all angles formed on the same side of a line at a given point on the line is 180 degree.

7. The sum of all angles around a point is 360.

8. Two angles are called a pair of vertically opposite angles if their arms form two pairs of a opposite rays.

9. If two lines intersect each other then the vertically opposite angles are equal.

Some Examples based on complement and supplement of angles

Example 1.
Find the complement of each of the following angles:
i) 50 degree
ii) 30 degree
iii) 78 degree

Solution

i) The given angle is 50 degree.
Let the measure of its complement be x degree. Then,
x 50 = 90 or x = 90 - 50 = 40.
Hence, the complement of the given angle measures 40 degree.

ii) The given angle is 30 degree. Let the measure of its complement be x degree. Then,x 30 = 90 or x = 90 - 30 = 60.Hence, the complement of the given angle measures 60 degree.

iii) The given angle is 78 degree. Let the measure of its complement be x degree. Then,x 78 = 90 or x = 90 - 78 = 12.Hence, the complement of the given angle measures 12 degree.

Example 2

Find the supplement of each of the following angles:
i) 120 degree
ii) 65 degree
iii) 48 degree

Solution

i) The given angle is 120 degree. Let the measure of its supplement be x degree. Then,x 120 = 180 or x = 180 - 120 = 60.
Hence, the supplement of the given angle measures 60 degree.

ii) The given angle is 65 degree. Let the measure of its supplement be x degree. Then,x 65 = 180 or x = 180 - 65 = 115.
Hence, the supplement of the given angle measures 115 degree.

iii) The given angles is 48 degree. Let the measure of its supplement be x degree. Then,x 48 = 180 or x = 180 - 48 = 132.
Hence, the supplement of the given angle measures 132 degree.

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By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

By using the principle of mathematical induction prove that 1.2.3+ 2.3.4+ 3.4.5 .... +n(n+ 1)(n+ 2) = n(n +1)(n+ 2)(n+ 3) /4

Solution

Let P(n) : 1.2.3 + 2.3.4 + 3.4.5 ... + n(n + 1)(n + 2) = n(n + 1)(n + 2)(n + 3) / 4

For n = 1
L. H. S. = 6 and R. H. S. = 6.
Therefore, L. H. S. = R. H. S. i.e., P(1) is true.
Let us suppose P(k) is true.
Therefore, putting n = k, we have,
1.2.3 + 2.3.4 + 3.4.5 .... +k(k + 1)(k + 2) = k(k + 1)(k + 2)(k + 3) / 4. ................................................(1)
Adding (k + 1)(k + 2)(k+ 3) to both sides of (1), we have,
1.2.3 + 2.3.4 + 3.4.5 ... + k(k + 1)(k + 2) + (k + 1)(k + 2)(k + 3) = {k(k + 1)(k + 2)(k + 3) /4}  + (k 1)(k 2)(k 3).
                                                                                                     =(k + 1)(k + 2)(k + 3) {(k/4) + 1}.
                                                                                                     =(k + 1)(k + 2)(k + 3)(k + 4) / 4.
                                                                                                    =(k+ 1)(k+ 1+ 1)(k +1 +2)(k+ 1+ 3) / 4.
Here P(n) is true for n = k+ 1 i.e. P(k+ 1) is true.
Therefore, By the principle of mathematical induction, P(n) is true for all natural numbers n.

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