Alternate Angles

When two lines are crossed by transversal then two types of alternate angles are produced.
⟹ Alternate Interior Angle
⟹ Alternate Exterior Angle

In the below figure;
M & N are two parallel lines and they are cut by transversal T

Alternate Interior Angle
These set of angles is on the interior of the parallel lines (MN) and are opposite to each other.

In the below figure:
\angle 4\ and\ \angle 6 are alternate interior angle
\angle 3\ and\ \angle 5 are alternate interior angle

Alternate Exterior Angle
These set of angles are on the exterior part of the lines and are opposite side of transversal

In the below figure:
\angle 1\ and\ \angle 7 are alternate exterior angle
\angle 2\ and\ \angle 8 are alternate exterior angle

Alternate Angles when lines are parallel

In the case when the two parallel lines are cut by transversal, then all set of alternate angles are equal.
⟹ alternate interior angle are equal
⟹ alternate exterior angle are equal

Equal set of Alternate Interior Angles are:
\angle 4\ =\ \angle 6
\angle 3\ =\ \angle 5

Equal Set of Alternate Exterior Angles are:
\angle 1\ =\ \angle 7
\angle 2\ =\ \angle 8

Note: Remember the set will be equal only if the lines are parallel to each other

Property of Alternate Angles

(a) Alternate Angle pair lie on opposite side of the transversal

you can observe in the above figure that:
\angle 4\ and\ \angle 6 pair of alternate angle are on the opposite side

(b) Alternate Angles Pair are either internal angles or external angles

Interior Angle: Angle that are inside the horizontal lines
External Angle: Angle that are outside the horizontal lines


Alternate Interior Angle consist of just interior angles of the given lines
Example:
\angle 4\ and\ \angle 6
\angle 3\ and\ \angle 5

Both pairs involve angle just in the interior parts

Alternate Exterior Angles involve angle pairs located just in the exterior parts
\angle 1\ and\ \angle 7
\angle 2\ and\ \angle 8

(c) Alternate Angle Pairs are equal when the lines are parallel

(d) Supplementary Alternate Angles

Two angles are said to be supplementary when they add up to number 180 degree.

Alternate Angle can be supplementary when the parallel lines are crossed by transversal at 90 degrees.

In the above figure;
M & N are parallel lines intersected by transversal T at 90 degree angle
\angle 4\ =\ \angle 6\ =\ 90 \angle 4\ + \angle 6\ = 180 degree

Alternate Angles Theorem Proof

Theorem
When two parallel lines are intersected by a transversal, then the alternate interior angles are equal

To Prove
\angle c\ = \angle e\

Solution
M & N are two parallel lines intersected by transversal T

As M is a straight line;
we know that sum of angles in a straight line is 180 degree
\angle a\ + \angle b\ = 180 degree —-eq(1)

Similarly
Transversal T is a straight line
\angle b\ + \angle c\ = 180 degree —-eq(2)

On Comparing equation (1) & (2), we concluded that:
\angle a\ = \angle c\ —–eq (3)

Using the concept of corresponding angles, we know that;
\angle a\ = \angle e\ —–eq (4)

Using eq(3) & eq(4), we conclude that:
\angle c\ = \angle e\

Hence proved that when lines are parallel. alternate interior angles are equal

FAQ’s – Alternate Angles

(01) Are Alternate Angles equal?

Read Solution

(02) How to remember alternate angles easily?

Just remember two points
(a) The pairs are opposite to each other
(b) The angles are either internal or external angle

(03) Can alternate angle be supplementary?

Read Solution

(04) How many types of alternate angles are there?

Alternate Angle Questions

(01) Fine the value of angle x in he below figure

(02) Find the value of angle y in the below image

(03) Find the value of angle c in below image

Since OP is a straight line, the sum of angle will be 180 degree

⟹ \angle a\ + \angle b\ = 180 degree
⟹ 71 + \angle b\ = 180
⟹ \angle b\ = 180 – 71
⟹ \angle b\ = 109

Since both the given lines are parallel.
\angle b\ = \angle c\ {Alternate Exterior Angle}
\angle c\ = 109 degree

(04) Find the value of angle c