## Angles

Subject: Compulsory Maths

#### Overview

The amount of turn between two straight lines that have a common endpoint that is a vertex is called angle. This note includes the pairs of angles made by a transversal with lines and different Pairs of angles.

#### Angles

The amount of turn between two straight lines that have a common endpoint that is a vertex.

Different Pairs of angles

$\angle$AOB and$\angle$BOC have common vertex O and a common arm OB. They are called adjacent angles.

2. Linear Pair

$\angle$AOB and$\angle$BOC are a pair of adjacent angles. Their sum is a straight angle (180°).
i.e$\angle$AOB +$\angle$BOC = 180°
$\angle$AOB and$\angle$BOC are called a linear pair.

3. Vertically Opposite Angles

$\angle$AOC and$\angle$BOD are formed by the intersected line segments and they lie to the opposite side of the common vertex. They are called vertically opposite angles.$\angle$AOD and$\angle$BOC are another pairs of vertically opposite angles. Vertically opposite angles are always equal.
$\therefore$$\angle$AOC =$\angle$BOD and$\angle$AOD =$\angle$BOC.

4. Complementary Angles

The sum of$\angle$AOB and$\angle$BOC is a right angle (90°). i.e$\angle$AOB +$\angle$BOC = 90°
$\angle$AOB and$\angle$BOC are called complementary angles.
Here, complement of$\angle$AOB = 90° -$\angle$BOC
Complement of$\angle$BOC = 90° - AOB

5. Supplementary Angles

The sum of $\angle$AOB and$\angle$BOC is two right angles (180°).
i.e$\angle$AOB +$\angle$BOC = 180°
$\angle$AOB and$\angle$BOC are called supplementary angles.
Here, supplement of$\angle$AOB = 180° -$\angle$BOC
Supplement of$\angle$BOC = 180° -$\angle$AOB

Pairs of angles made by a transversal with lines

In the given figure, AB and CD and two parallel lines (AB//CD). PQ is the transversal that intersects AB at R and CD at S.

1. Exterior and alternate exterior angles

$\angle$a,$\angle$b,$\angle$c and$\angle$d are lying outside the parallel lines. They are called exterior angles.$\angle$a and$\angle$d are lying to the opposite side of the transversal. They are called alternate exterior angles.
The alternate exterior angles made by a transversal with parallel lines are always equal.
$\therefore$$\angle$a =$\angle$d and$\angle$b =$\angle$c
2. Interior and Co-interior angles

$\angle$a,$\angle$b,$\angle$c and$\angle$d are lying inside the parallel lines. They are called interior angles. $\angle$a and$\angle$c are the pair of interior angles lying to the same side of the transversal. They are called Co-interior angles. The sum of a pair of co-interior angles made by a transversal with parallel lines is always 180°.
$\therefore$$\angle$a + $\angle$c = 180° and$\angle$b +$\angle$d = 180°
3. Alternate angles

$\angle$a and$\angle$d are a pair of interior angles lying to the opposite side of a transversal and they are not adjacent to each other. They are called alternate angles.$\angle$b and$\angle$c are another pairs of alternate angles.
A pair of alternate angles made by a transversal with parallel lines is always equal.
$\therefore$$\angle$a =$\angle$d and$\angle$b =$\angle$c

4. Corresponding angles

$\angle$a is an exterior and$\angle$d is an interior angle lying to the same side of the transversal and they are not adjacent to each other.; They are called corresponding angles. b and d are another pairs of corresponding angles.
A pair of corresponding angles made by a transversal with parallel lines is always equal.
##### Things to remember
• Definition of angles
• Different pairs of angle
• Pairs of an angle made by transversal lines.
• It includes every relationship which established among the people.
• There can be more than one community in a society. Community smaller than society.
• It is a network of social relationships which cannot see or touched.
• common interests and common objectives are not necessary for society.
##### Types of angles

Solution:

Here, x° + (x+10)° = 90° [The sum of a pair of complementary angles]

or, 2x° = 90° - 10°

or, x° = $\frac{80°}{2}$ = 40°

$\therefore$ x° = 40° and (x+10)° = 40° + 10° = 50°

Solution:

Let the required supplementary angles be 3x° and 2x°.

$\therefore$ 3x° + 2x° = 180° [The sum of a pair of supplementary angles]

or, 5x° = 180°

x° = $\frac{180°}{5}$ = 36°

$\therefore$ 3x° = 3 × 36° = 108°

2x° = 2 × 36° = 72°

Solution:

Here, x° + 2x° + 3x° = 180° [Being the sum a straight angle]

or, 6x° = 180°

or, x° = $\frac{180°}{6}$ = 30°

$\therefore$ x° = 30°, 2x° = 2×30° = 60° and 3x° = 3×30° = 90°

Again, a° = x° = 30°, b = 2x° = 60° and c° = 3x° = 90° [Each pair is vertically opposite angles]

Solution:

w = 110° [Being vertically opposite angles]

x = w = 110° [Being alternate angles]

y = x = 110° [Being vertically opposite angles]

y + z = 180° [Being the sum of a pair of co-interior angles]

or, 110 + z = 180°

or, z = 180° - 110° = 70°

So, w = x= y = 110° and z = 70°

The different pairs of angles are: