 ## Angles

Subject: Compulsory Maths

#### Overview

An angle is a figure formed by two rays, called the sides of the angle sharing a common end point.The types of angles are classified according to their measure. The angles are; adjacent angles, complementary angles, supplementary angles, vertical angles etc.

##### Angles

Measurement of angles discriminate the types of angles. Through measurement, we have pairs of angles likewise adjacent angles, complementary angles, supplementary angles, vertical angles, etc. In the above figure, we can observe angle having a common side and common vertex and they do not overlap. Hence, the given two angles are adjacent.

In next figure, two angles are not adjacent angles even they have common sides and vertex as they overlap.

### Complementary Angles

If two angles make up 90o by adding up then it is called complementary angles. But angles do not have to be together. Let’s us know more with figure help. In the given figure, two angles (40 and 50) are complementary angles because they add up to 90o.   These two angles are complementary because 27° + 63° =90o

### Supplementary Angles

If two angles make up to 180o by adding then it is called supplementary angles. But angles do not have to be together. Let’s us know more with figure help. In the given figure, two angles (60o and 120o) are complementary angles because they add up to 180o. These two angles are supplementary because 60° + 120° = 180°.

### Vertical Angles

Angles sharing the same vertex are called vertical angles. Angles share their vertex when two line intersect and it form vertical angles or vertically opposite angles. In the given figure, $\angle$p and $\angle$s are opposite to $\angle$r and $\angle$q. Therefore $\angle$p and $\angle$r are vertical angles. Similarly $\angle$s and$\angle$ q are also vertical angles.

### Vertical Angle Theorem

Experiment:

In the given figure, two lines PQ and RS intersect each other. Look at the figure and complete the table given below. Statements Reasons 1. a + c =180° 1. Supplementary angles 2. b + c =180° 2. ................... 3.a + c ≅ b + c 3. From 1 and 2 4. a ≅ b 4. Concealing c from both sides 5. Similarly, c ≅ d 5. As above

### Parallel lines

Two lines lie in the same plane but do not intersect each other is known as parallel lines.

In the given figure, the lines AB and CD are parallel. Mathematically,AB//CD. Parallel lines are apart always at the same distance .Hence, the distance between two parallel lines is the same everywhere. In the figure, PQ//RS.

Take any point M on PQ and draw MN$\perp$RS.

Take any point E on PQ and draw EF$\perp$RS.

Measure MN and EF.

You will find that MN=EF.

### Transversal

The line that intersects two or more lines is called transversal. Transversal lines create an angle where some them have a name and give relation to the lines.

Let's identify some of the angles with name and their relation.

i) a and b are alternative interior angles. ii) a and b are alternative exterior angles. iii) a and b are corresponding angles. iv) a and b are interior angles on the same side. ### Alternative Angles Theorem

When transversal cuts two line forming corresponding angles which are congruent prove two line to be parallel.

Experiment:

Draw a pair of Line AB and CD which are parallel and intersect them by transversal EF. Name the interior angles as 3, 4, 5 and 6 and exterior angles as 1, 2, 7 and 8. In the given figure AB//CD. Complete the table below.

 S.No. Statements Reasons 1. 3 ≅ 2 Vertical angles 2. 2 ≅ 6 Corresponding angles 3. 3 ≅ 6 Transitive property 4. 5 ≅ 8 .................... 5. 5 ≅ 4 .................... 6. 1 ≅ 4 .................... 7. 3 ≅ 7 ....................

### Alternative Angles Converse

Alternative angles converse proves the two line are parallel by cutting the lines by a transversal and forming a congruent angle. In the given figure x$\cong$c. Complete the table.

 S.No. Statements Reasons 1. x ≅ c Given 2. b ≅ c Vertical angles 3. x ≅ b ....................... 4. b ≅ z ....................... 5. b ≅ w ....................... 6. d ≅ y ....................... 7. PQ//RS .......................

### Corresponding Angles Theorem

Corresponding angles are equal in parallel line which is formed by transversal cutting the parallel line. In the figure, PQ//RS

So, $\angle$1 $\cong$ $\angle$5, $\angle$7 $\cong$ $\angle$3, $\angle$8 $\cong$ $\angle$4 and $\angle$2 $\cong$ $\angle$6

### Corresponding Angles Converse

When corresponding angles are congruent drawing the transversal line then the crossed line by a transversal are parallel. In the figure, $\angle$a$\cong$$\angle$b, so PQ//RS.

### Consecutive Interior Angles Theorem

Consecutive interior angles theorem states that consecutive interior angles form by two parallel lines and a transversal are supplementary.

In the figure, PQ//RS. Look at the figure and complete the table below: S.No. Statements Reasons 1. b+m=180o Being supplementary angles 2. m$\cong$c ............... 3. b+c=180o ............... 4. Similarly, a+d=180o ...............

### Consecutive Interior Angles Converse

If transversal forms interior angles that are supplementary angles by cutting two line, then the lines are parallel.

In the figure, m+y=180o Complete the table that is given below.

 S.No. Statements Reasons 1. m+y=180o Supplementary angles 2. n+y=180o .............. 3. m+y$\cong$n+y .............. 4. m$\cong$n .............. 5. PQ//RS ..............

##### Things to remember
• When two rays meet, they create an angle.
• Angles are measured with the tool called a protector.
• The red and black crossbar should be lined up with the vertex of the angle.
• The vertex is the point where two rays of an angle meet.
• 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.

Solution:

Here, x+3x=180? [Sum of adjacent angles]

or, 4x=180?

or, x=$\frac{180?}{4}$=45?

? x= 45?

Solution,

Here, x=45? [ Vertically opposite angle ]

?x=45?

Again, x+y=180?

or, 45?+y=180?

or, y=180?-45?

or, y=135?

$\therefore$y=135?

Solution:

Here, x+100?=180?

or, x=180?-100?

or, x=80?

$\therefore$ x=80?

Again, y=x [vertically opposite angle]

or, y=80?

Solution:

Here, x=60? [ ?Vertically opposite angle]

?x=60?

Again, y=70?[?Vertically opposite angle]

?y=70?

Solution,

Here, x+45?=180? [?Sum of Adjacent Angles]

or, x=180?-45?

or, x=135?

Again, y=45?[?vertically opposite angle]

?y=45?

and z=x [?vertically opposite angle]

or, z=135?

?z=135?

then, x=135?,y=45? and x=135?

Here, x=50?[?vertically opposite angle]

?x=50?

Again, 50? + z + 50?=180?[Sum of adjacent angle in a sameline]

or, z=180?-100?

or, z=80?

$\therefore$z=80?

and, y=z

or, y=80?

$\therefore$y=80?

So, x=50?, y=80? and z=80?

Solution:

Step 1: x is a supplement of 65?

Therefore, x+65?=180?

x=180?-65?=115?

Step 2: x and 115? are vertically opposite angles.

Therefore, x = 115?

Step 3: y and 65? are vertically opposite angles

Therefore, y=65?

$\therefore$ x=115?, y=65? and z=115?

Solution:

a and c are supplementary angles.

Therefore, a+c=180°

150°+c=180°

c=180°-150°=30°

b and c are vertical angles.

Therefore, b=c=30°

a and d are corresponding angles.

Therefore, a=d=150°

c and e are corresponding angles.

Therefore, c=e=30°

Solution:

a+50?+30?=180?(sum of adjacent angles formed on a straight line)

a+80?=180?

a=180?-80?=100? and d=30?(being alternative angles)

b+50?=180?(Sum of consecutive interior angles)

b=180?-50?=150?

e+30?=180?(sum of consecutive interior angles)

e=180?-30?=150?

Solution:

b+158?=180? (supplementary angles)

b=180?-158?=22?

c=37? ( alternative angles)

Again, a+37?=158? (supplementary angles)

a=158?-37?=121?

Solution:

Here, y=490[vertically opposite angles]

Again, x=y [alternative angles]

or, x=490

So, x=490 and y=490

Solution:

Here, x=115? [ Vertically opposite angle]

Likewise, y=x=115?[ Corresponding angles]

Again, z=58? [Vertically opposite angles]

and a=z=58? [ Alternate angle]

$\therefore$ a=58?

So, a=58?, x=115?,z=58?

Solution:

Here, z+75?=180? [$\therefore$adjacent angles]

or, z=18?-75?=105?

$\therefore$ z=105?

Again, x=z=180? [$\therefore$Sum of co-interior angles]

or, x+105?=180?

or, x=180?-105?=75?

Likewise, y=z [$\therefore$ corresponding angles ]

? y=105?

and a=x [ corresponding angles]

So, a=75?,x=75?, y=105?,z=105?

Solution:
Convert 2/3 of 90°
2/3 × 90° = 60°
Complement of 60° = 90° - 60° = 30°
Therefore, complement of the angle 2/3 of 90° = 30°

Solution:
Convert 4/5 of 90°
4/5 × 90° = 72°
Supplement of 72° = 180° - 72° = 108°
Therefore, supplement of the angle 4/5 of 90° = 108°

Solution:
According to the question, (2x - 7)° and (x + 4)°, are complementary angles’ so we get;
(2x - 7)° + (x + 4)° = 90°
or, 2x - 7° + x + 4° = 90°
or, 2x + x - 7° + 4° = 90°
or, 3x - 3° = 90°
or, 3x - 3° + 3° = 90° + 3°
or, 3x = 93°
or, x = 93°/3°
or, x = 31°
Therefore, the value of x = 31°.

Solution:
According to the question, (3x + 15)° and (2x + 5)°, are complementary angles’ so,
(3x + 15)° + (2x + 5)° = 180°
or, 3x + 15° + 2x + 5° = 180°
or, 3x + 2x + 15° + 5° = 180°
or, 5x + 20° = 180°
or, 5x + 20° - 20° = 180° - 20°
or, 5x = 160°
or, x = 160°/5°
or, x = 32°
Therefore, the value of x = 32°.