# Lines and their Properties

A line is a set of points placed together that extends into infinity is both directions.

# Different angles and pairs of angles

Measurement and nomenclature**Acute Angle**

Property : 0Â° <Î¸< 90Â° (âˆ AOB is an acute angle)

**Right Angle**

Property : Î¸ = 90Â°

**Obtuse Angle**

**Straight Line**

**Reflex Angle**

**Complementary Angle**

_{1}+ Î¸

_{2}= 90Â°

Two angles whose sum is 90Â° are complementary to each other

**Supplementary Angle**

Property: Î¸1 + Î¸2 = 180Â°

Two angles, whose sum is 180Â°, are supplementary to each other

**Vertically Opposite Angle**

**Adjacent Angles**

**Linear Pair**

**Angles on One Side of a Line**

_{1 }+Î¸

_{2 }+Î¸

_{3 }= 180Â°

**Angles Round the Point**

_{1 }+ Î¸

_{2 }+ Î¸

_{3 }+ Î¸

_{4 }= 360Â°

**Angle Bisector**

(Angle bisector is equdistant from the two sides of the angle) i.e., |

Property: OC is the angle bisector ofâˆ AOB. i.e., âˆ AOC = âˆ BOC (âˆ AOB) When a line segment divides an angleequally into two parts, then it is said to be the angle bisector (OC)

# Angles associated with two or more straight lines

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When two straight lines cross each other, âˆ d and âˆ b are the pair of vertical angles.

âˆ a and âˆ c are the pair of vertical angles.

Vertical angles are equal in value.

*Alternate angles and corresponding angles*

In the figure given below, corresponding angles are âˆ a and âˆ e, âˆ b and âˆ f, âˆ d and âˆ h, âˆ c and âˆ g. The alternate angles are âˆ b and âˆ h, âˆ c and âˆ e.

**Corresponding angles**

When two lines are intersected by a transversal, then they form four pairs of corresponding angles

- âˆ AGE, âˆ CHG = (âˆ 2, âˆ 6)
- âˆ AGH, âˆ CHF = (âˆ 3, âˆ 7)
- âˆ EGB, âˆ GHD = (âˆ 1, âˆ 5)
- âˆ BGH, âˆ DHF = (âˆ 4, âˆ 8)

# Angles associated with parallel lines

A line passing through two or more lines in a plane is called a transversal. When a transversal cuts two parallel lines, then the set of all the corresponding angels will be equal and similarly, the set of all the alternate angles will be equal.In the figure given above, corresponding âˆ a = âˆ e and corresponding âˆ b = âˆ f

Similarly, alternate âˆ b = âˆ h and alternate âˆ c = âˆ e.

Now, âˆ b + âˆ c = 180Â°, so âˆ b + âˆ e = âˆ h + âˆ c = 180Â°

So we can conclude that the sum of the angles on one side of the transversal and between the parallel lines will be equal to 180Â°.

Converse of the above theorem is also true. When a transversal cuts two lines, and if the corresponding angles are equal in size, or if alternate angles are equal in size, then the two lines are parallel.

Example-1

In the figure given below, find the value of âˆ b in terms of âˆ a.

Solution

In the given figure, âˆ b = Alternate âˆ PDC = 180Â°

^{ }âˆ’âˆ PDA = 180Â°^{ }âˆ’ a