Basics of angles, their measures and various types of angles are explained in the chapter Angles And Its Types Based On Its Different Measures. It says angles are formed only when two lines or rays intersect at some point.
Also, angles are formed when a transversal line intersects two or more parallel lines. Such angles formed can lie inside or outside of the parallel lines and are named as interior and exterior angles respectively.
Exterior angles
Exterior angles are formed at points on the exterior sides of the two lines where a transversal line intersects these two lines.
∠1, ∠2, ∠7 and ∠8 are formed at points A and B on the exterior sides of two parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, ∠1, ∠2, ∠7 and ∠8 are called exterior angles.
Interior angles
Interior angles are formed at points on the interior sides of the two lines where a transversal line intersects these two lines.
∠3, ∠4, ∠5 and ∠6 are formed at points A and B on the interior sides of two parallel lines l and m respectively, when a transversal line n cuts through them. Therefore, ∠3, ∠4, ∠5 and ∠6 are called interior angles.
Transversal crosses parallel lines
When a transversal crosses parallel lines it creates specific pairs of angles which are always equal. These equal pair of angles formed are pair of alternate angles or pair of corresponding angles.
Alternate angles
Alternate angles again are formed when a transversal line cuts through two parallel lines. These angles are a pair of angles which can exist on the interior or exterior sides of the two lines.
Here, a transversal line n cuts through two lines l and m and forms total of eight angles at points A and B which are: ∠1, ∠2, ∠3, ∠4, ∠5, ∠6, ∠7 and ∠8.
If we select the pairs of angles as (∠1 and ∠7), (∠2 and ∠8), (∠3 and ∠6) and (∠4 and ∠5), then these pairs are the alternate angles.
Depending upon where the pair exists on the two lines l and m we can name them alternate interior angles and alternate exterior angles.
Alternate exterior angles
In the above diagram of alternate angles, the pairs of alternate angles ∠1 and ∠7, ∠2 and ∠8 lie outside the lines i.e. exterior sides of the two lines, such type of alternate angles are called alternate exterior angles.
Alternate interior angles
Similarly, the pairs of alternate angles ∠3 and ∠6, ∠4 and ∠5 lie inside the lines i.e. interior sides of the two lines, such type of alternate angles are called alternate interior angles.
These pairs of alternate exterior and interior angles are always equal.
So, we can say that the following alternate exterior angles pairs are equal
- ∠1 = ∠7
- ∠2 = ∠8
Also, the alternate interior angles pairs are equal
- ∠3 = ∠6
- ∠4 = ∠5
Corresponding angles
A pair of angles is called corresponding angles in which one arm of both angles is on the same side of transversal and their other arms are directed in the same sense.
So, from the above diagram of corresponding angles, the following pair of angles are corresponding angles that are formed on two parallel lines l and m.
- ∠1 and ∠6
- ∠2 and ∠5
- ∠3 and ∠7
- ∠4 and ∠8
If lines l and m are parallel, then pairs of corresponding angles are always equal. Therefore, from the same above example of corresponding angles we can write them as:
- ∠1 = ∠6
- ∠2 = ∠5
- ∠3 = ∠7
- ∠4 = ∠8
In parallel lines l and m, sum of interior angles of the same side of transversal is 180°
From the above figure of corresponding angles, ∠4 and ∠6 are interior angles on the same side of transversal line n
- ∴ ∠4 + ∠6 = 180°
- Similarly, ∠3 + ∠5 = 180°
Linear pair of angles
When the sum of two adjacent angles is 180°, they are called linear pair of angles. Or, we can say when supplementary angles formed on a straight line, those angles are called a linear pair of angles.
Here, ∠ABO + ∠CBO = 180°, therefore, they form a linear pair of angles.
In the above figure of corresponding angles, the following pairs of angles form the linear pairs of angles:
- ∠1 and ∠2
- ∠1 and ∠4
- ∠2 and ∠3
- ∠3 and ∠4
- ∠5 and ∠6
- ∠6 and ∠8
- ∠5 and ∠7
- ∠7 and ∠8
Vertically opposite angles
Vertically opposite angles are not formed by a transversal and parallel lines but when two line intersect each other.
When two straight lines intersect each other, they form four angles at the point of intersection. Out of four angles the two angles which are directly opposite to each other are called vertically opposite angles. This pair of vertically opposite angles are always equal.
Here, in the above diagram, we can see ∠AOD, ∠BOC, ∠AOC and ∠BOD are the four angles formed at point O when two lines AB and CD intersect at point O.
The angles ∠AOD and ∠BOC are directly opposite to each other, therefore they are called vertically opposite angles and ∠AOD = ∠BOC
Similarly, the angles ∠AOC and ∠BOD are also directly opposite to each other, therefore they are also called vertically opposite angles and ∠AOC = ∠BOD
Moreover, the sum of each pair of adjacent angles is always equal to 180°.
- ∠AOC + ∠BOC = 180°
- ∠AOC + ∠AOD = 180°
- ∠BOC + ∠BOD = 180°
- ∠AOD + ∠BOD = 180°
