Straight Line

Angle

Consider 2 straight lines AB, CD intersecting at O

AO can be rotated about O so that it lies upon CO, occupying the same set of points

The amount of revolution is known as the angle AOC

If AO continues to rotate about O until it once again overlays its original position it will have performed 1 revolution.

One measure divides 1 revolution into 360 degrees, each degree into 60 minutes and each minute into 6o seconds i.e.

1 revolution = 360 degrees = 2,160 minutes = 1,296,000 seconds

(5º is shorthand way of writing 5 degrees)

Right Angles

CD is a straight line.

O is a position that lies on CD and AO is another straight line.

∠AOD is the amount a line OD needs to be rotated to position it over OA.

    (“∠” is the symbol for angle)

 All values for this angle are possible, including the one that satisfies 

   ∠AOD = ∠AOC 

This angle is called a right angle (rt. ang.) and AD is said to be perpendicular to CD

If one rotates AO another 1 rt. ang. anti-clockwise it lies on CO

   since ∠AOD = ∠AOC = 1 rt. ang

This is equivalent to rotating OD through 2 rt. ang.s.

But CO is part of straight line CD, so a straight line may be thought of the addition of 2 rt. ang.s.

BO is drawn such that ∠COB = ∠BOD then as before these are rt. ang.s

then as AB is also built of 2 rt. angs. it is also a straight line.

So 4 rt. ang.s = 1 revolution = 360º   ∴ 1 rt. ang. = 90º

(Currently we have shown how a rt. ang. can exist, later it will be shown how to construct one) 

Opposite angles of intersecting lines are equal

AB & CD are two straight lines intersecting at O.

Let MN be a line passing through O perpendicular to AB

∴ ∠MOB = ∠NOB =  ∠AON = ∠AOM = 1 rt. ang. 

  ∠DOB = 1 rt. ang. - ∠NOD  

  ∠AOC = 2 rt. ang.s - ∠AOD 

       = 2 rt. ang.s - 1 rt. ang. - ∠NOD 

       = 1 rt. ang. - ∠NOD 

       = ∠DOB

Similalry ∠DOA = ∠COB AB & CD are two straight lines intersecting at O.

Let MN be a line passing through O perpendicular to AB

∴ ∠MOB = ∠NOB =  ∠AON = ∠AOM = 1 rt. ang. 

  ∠DOB = 1 rt. ang. - ∠NOD  

  ∠AOC = 2 rt. ang.s - ∠AOD 

       = 2 rt. ang.s - 1 rt. ang. - ∠NOD 

       = 1 rt. ang. - ∠NOD 

       = ∠DOB

Similalry ∠DOA = ∠COB