Introduction

In geometry, lines are the figures used to make infinite points and can be extended infinitely in both the directions. Lines are always straight and contains negligible depth as well as width. There is a variety of lines like perpendicular lines, intersecting lines, transversal lines, etc. Lines are divided into two types as line segments and rays. 

• The line segment is a part of the line which has two endpoints. It has a fixed and shortest distance between the endpoints. 

• The ray is also a part of the line with its starting point but no ending point; that is, lines will be extended infinitely in one direction.

An angle is formed when the two lines or say ray emerge from each other at any common point. There are different angles: acute angle, right angle, obtuse angle, straight angle and reflex angle. 

• The acute angle is the angle that is less than $$90^{\circ}$$ but greater than $$0^{\circ}$$. 

• The right angle is the angle that is always equal to $$90^{\circ}$$. 

• The obtuse angle is the angle that is greater than $$90^{\circ}$$ and less than $$180^{\circ}$$. 

• The straight angle is the angle that is always equal to $$180^{\circ}$$.

• And the reflex angle is the angle that is greater than $$180^{\circ}$$ and less than $$360^{\circ}$$.

E4.7A: Calculate unknown angles using the following geometrical property angles at a point.

If the sum of the two angles is $$90^{\circ}$$ then they are known as the complementary angles, and if the sum of the two angles is equal to $$180^{\circ}$$ then they are known as the supplementary angles. The angles that contain the common sides and common vertices are known as the adjacent angles. 

Vertically opposite angles are two angles formed opposite to each other when the lines intersect at a common point or vertex. In the figure, by using vertically opposite angles $$\angle{a}=\angle{b}$$ and $$\angle{c}=\angle{d}$$.

Worked examples:

Example 1: As shown in the figure below, $$AB$$ and $$CD$$ are two lines that intersect each other at point $$O$$. If the ratio of $$\angle{AOD}$$ and $$\angle{DOB}$$ is $$4:5$$. Find all the angles.

Step 1: State the given information.

$$\angle{AOD}:\angle{DOB}=4:5$$

Step 2: By applying the linear pair of angles

$$\angle{AOD}+\angle{DOB}=180^{\circ}$$

Step 3: Find the angle $$AOD$$.

$$\angle{AOD}=\frac{4}{9}\times{180}^{\circ}$$. So, value of $$\angle{AOD}=80^{\circ}$$.

Step 4: Find the angle $$DOB$$.

$$\angle{DOB}=\frac{5}{9}\times{180}^{\circ}$$. So, value of $$\angle{DOB}=100^{\circ}$$.

Step 5: Find the angle $$AOC$$.

$$\angle{AOC}=100^{\circ}$$ because $$\angle{DOB}$$ and $$\angle{AOC}$$ are vertically opposite angles.

Step 6: Find the angle $$COB$$.

$$\angle{COB}=80^{\circ}$$ because $$\angle{AOD}$$ and $$\angle{COB}$$ are vertically opposite angles.

So, the value of angles are $$\angle{AOD}=80^{\circ}$$, $$\angle{DOB}=100^{\circ}$$, $$\angle{AOC}=100^{\circ}$$ and $$\angle{COB}=80^{\circ}$$.

E4.7B: Calculate unknown angles using the following geometrical property angles at a point on a straight line and intersecting straight lines.

There are different types of lines, such as intersecting lines, parallel lines, and transversal lines. Here, we are going to discuss the intersecting lines. Intersecting lines are the two lines that intersect each other at a point.

One of the properties of angles is: the sum of the angles at one side of a straight line are always equal to $$108^{circ}$$, whereas the sum of the angle around a given point is always equal to the $$360^{circ}$$.

There is a theorem for calculating the angles of the two-intersecting lines. Theorems are if there are two intersecting lines, then the vertically opposite angles formed will be equal.

Worked examples:

Example 1: In the figure below, $$PQ$$, $$RS$$ and $$TO$$ are lines which intersect each other at point $$O$$. Two angles are given $$\angle{SOQ}=100^{circ}$$ and $$\angle{ROT}=45^{circ}$$. Find the value of $$\angle{POT}$$ and  $$\angle{ROQ}$$.

Step 1: State the given information.

$$\angle{POT}=a$$, $$\angle{ROQ}=b$$, $$\angle{SOQ}=100^{circ}$$, and $$\angle{TOR}=45^{\circ}$$.

Step 2: Find the angle $$b$$.

$$\angle{SOQ}=100^{\circ}$$ and it is observed that $$\angle{SOQ}+\angle{ROQ}=180^{\circ}$$ as sum of angles on a straight line is $$180^{\circ}$$.

Step 3: Substitute the value of $$\angle{SOQ}=100^{circ}$$ in $$\angle{SOQ} + \angle{ROQ} =180^{circ}$$.

$$100^{circ}+b=180^{circ}$$. So, value of $$\angle{ROQ}=80^{\circ}$$.

Step 4: Find the angle $$a$$.

$$\angle{ROT}=45^{\circ}$$ and it is observed that $$\angle{POT}+\angle{TOR}+\angle{ROQ}=180^{\circ}$$ as the sum of angles on straight line is $$180^{\circ}$$.

Step 5: Substitute the value of $$\angle{ROT}=45^{circ}$$ and $$\angle{ROQ}=80^{circ}$$ in $$\angle{POT}+\angle{TOR}+\angle{ROQ}=180^{circ}$$.

$$a+45^{\circ}+80^{\circ}=180^{\circ}$$. So, value of $$a=55^{\circ}$$.

So, the value of $$\angle{ROQ}=80^{\circ}$$ and $$\angle{POT}=55^{\circ}$$.