Geometry Concepts: Lines and Angles

In the last post we have discussed points, planes and lines and some important things related to them. Today in this article we will move one step further and study the ANGLES made with the help of lines, types of angles, properties of angles and a theorem of parallel lines with a transversal. 

Angle

An angle is formed when two rays originate from the same point. This point is called the vertex of the angle and the originated rays are called the arms of the angle. The  symbol of angle is ''. 

For example this is ∠AOB in which O is the vertex and OA & OB are the arms. 

An angle can also be defined as the amount of rotation of a ray about its vertex. For example in the following figure, the ray OA changed its direction by rotating on its vertex O and now the new direction is OB. So, it makes ∠AOB. 

Another important thing ∠AOB and ∠BOA are the same thing. 
Types of angles

I have categorized all the angles under 3 heads. They are
  1. Angles based on measurement 
  2. Angles based on lines 
  3. Angles based on lines and a transversal 
Let's discuss them in detail.

Angles based on measurements
  1. Acute angle: whose value lies between 0o and 90o
  2. Right angle: whose value is 90o
  3. Obtuse angle: whose value lies between 90o and 180o
  4. Straight angle: whose value is 180o
  5. Reflex angle: whose value lies between 180o and 360o
  6. Complete angle: whose value is 360o
  7. Complementary angles: Two angles whose sum is 90o are complementary to each other. For eg. 30o and 60o are complementary to each other.
  8. Supplementary angles: Two angles whose sum is 180o are supplementary to each other. For eg. 40o and 140o are supplementary to each other.
Angles based on lines

1. Vertically Opposite angles: In the given figure,
  • ∠1 and ∠2
  • ∠3 and ∠4 
are the pair of vertically opposite angles and they are always equal. That means:
  • ∠1 = ∠2
  • ∠3 = ∠4
Vertically Opposite Angles
2. Adjacent angles: Two angles are adjacent if they have a
  • a common vertex, 
  • a common arm and 
  • their non-common arms are on different sides.  
Adjacent Angles
In the above figure ∠ABD and ∠DBC are adjacent angles because they have
  • a common vertex = B
  • a common arm = BD
  • their non-common arms are on different sides = AB and BC 
3. Linear Pair of angles: Now continuing the above example if BA and BC form a line then it will look like the following figure. In this case ∠ABD and ∠DBC are called the linear pair of angles.

Also, ∠ABD +∠DBC = 180o

The converse of this rule is also true that is, if two adjacent angles have sum equal to 180 degrees then they are called linear pair of angles.
Linear Pair of Angles

Angles based on lines and a transversal
1. Corresponding angles
  • ∠1 and ∠5
  • ∠3 and ∠7
  • ∠2 and ∠6
  • ∠4 and 8
2. Exterior angles: ∠1, ∠2, ∠7 and ∠8

3. Interior angles: ∠3, ∠4, ∠5 and ∠6

4. Alternate interior angles
  • ∠3 and ∠6
  • ∠4 and ∠5
Parallel lines with a transversal

Theorem 1 : When two lines are intersected by a transversal then following three results are always TRUE.
1. Corresponding angles are congruent / equal.
  • ∠1 = ∠5
  • ∠3 = ∠7
  • ∠2 = ∠6
  • ∠4 = 8
2. Alternate interior angles are congruent / equal.
  • ∠3 = ∠6
  • ∠4 = ∠5
3. Interior angles on the same side of the transversal are supplementary.
  • ∠4 + ∠6 = 180o
  • ∠3 + ∠5 = 180o
Theorem 2. (Converse of Theorem 1) :  The converse all the 3 rules is also true i.e. if the pair of 
  • corresponding angles or
  • alternate interior angles are equal or
  • interior angles on the same side of the transversal are supplementary 
then the two lines are parallel which are intersected by a transversal.

No comments:

Post a Comment

Hello, This is Vikas (your mentor) and this is a quick tutorial as how to post your comment.

You can use your gmail account to post your comments and in case you do not have a gmail account. Then you have two options. You can either choose 'Name/URL' option or 'Anonymous' option from 'Comment as:' menu.

If you are using 'Name/URL' option:
Just fill your name in the 'Name' box and hit 'Continue'.

And after that just type in your comment in the box provided and hit 'Publish'.

If you are using 'Anonymous' option:
In this case you will need to type your name in the comment box and then type your comment and hit 'Publish' button.

Just remember comment moderation is ON that means your comments will be published after our approval. So that only useful comments and content is available on the website.

Thank you and good luck for your exams :)