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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
Description: 22967.png

Property : 0° <θ< 90° (AOB is an acute angle)

Right Angle
Description: 22987.png

Property : θ = 90°

Obtuse Angle
Description: 22994.png
Property: 90° < θ < 180°

Straight Line
Description: 23007.png
Property: θ = 180°

Reflex Angle
Description: 23020.png
Property: 180° < θ < 360°

Complementary Angle
Description: 22980.png
Property: θ1 + θ2 = 90°
Two angles whose sum is 90° are complementary to each other

Supplementary Angle
Description: P-317-1.tif

Property: θ1 + θ2 = 180°
Two angles, whose sum is 180°, are supplementary to each other

Vertically Opposite Angle
Description: P-317-2.tif
Property: DOA = BOC and DOB =AOC

Adjacent Angles
Description: P-317-3.tif
Property: AOB and BOC are adjacent angles Adjacent angles must have a common side (e.g., OB)

Linear Pair
Description: P-317-4.tif
Property: AOB and BOC are linear pair angles. One side must be common (e.g., OB) and these two angles must be supplementary.

Angles on One Side of a Line
Description: P-317-5.tif
Property: θ1 2 3 = 180°

Angles Round the Point
Description: P-317-6.tif
Property: θ1 + θ2 + θ3 + θ4 = 360°

Angle Bisector
Description: P-317-7(1).tif Description: P-317-7(2).tif
(Angle bisector is equdistant from the two sides of the angle) i.e.,

Property: OC is the angle bisector ofAOB. i.e., AOC = BOC Description: 7585.png (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

Description: P-318-1.tif
When two straight lines cross each other, d and b are the pair of vertical angles.
a and 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 a and e, b and f, d and h, c and g. The alternate angles are b and h, c and e.
Description: 22893.png

Corresponding angles
Description: 22925.png

When two lines are intersected by a transversal, then they form four pairs of corresponding angles
  1. ∠AGE, ∠CHG = (∠2, ∠6)
  2. ∠AGH, ∠CHF = (∠3, ∠7)
  3. ∠EGB, ∠GHD = (∠1, ∠5)
  4. ∠BGH, ∠DHF = (∠4, ∠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.
Description: 23048.png
In the figure given above, corresponding a = e and corresponding b = f
Similarly, alternate b = h and alternate c = e.
Now, b + c = 180°, so b + e = h + c = 180°
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 180°.
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.
In the figure given below, find the value of ∠b in terms of ∠a.
Description: 22870.png
In the given figure, b = Alternate PDC = 180° PDA = 180° − a

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