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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.
 
Example-1
In the figure given below, find the value of ∠b in terms of ∠a.
 
Description: 22870.png
Solution
In the given figure, b = Alternate PDC = 180° PDA = 180° − a
 





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