There are two keywords to note here:
(a) Same Side
(b) Interior
So, when two lines are intersected by a transversal, then the angle formed on same side of the transversal and in the interior of the lines are known as Same Side Interior Angles.
Observe the image below:
M & N are two lines intersected by transversal T
Here;
Angles 1 & 2 are Same Side Interior Angles
Angles 3 & 4 are also Same Side Interior Angles
Same Side Interior Angle Theorem
The theorem says that when the two lines are parallel and are intersected by the transversal then the set of Same Side Interior Angles are Supplementary.
Important Point to note:
(a) Line should be parallel
(b) Same Side Interior Angles are Supplementary
In the above figure;
M & N are two parallel lines intersected by transversal T.
In this case, according to Same Side Interior Angle Theorem;
⟹ \angle 1\ + \ \angle 2 = 180
⟹ \angle 3\ + \ \angle 4 = 180
Proof of Same Side Interior Angle Theorem
Theorem
When two parallel lines are intersected by transversal, then the same side interior angles are supplementary
To Prove
⟹ \angle 1\ + \ \angle 2 = 180
Solution
Since transversal T is a straight line, the sum of angle in the line will add to 180 degree
⟹ \angle 1\ + \ \angle 3 = 180 — eq(1)
Since both line M & N are parallel, we can say that:
\angle 2\ = \angle 3\ {Corresponding Angle} —–eq(2)
Putting eq(2) in eq(1), we get:
⟹ \angle 1\ + \ \angle 2 = 180
Corollary of Same Side Interior Angle Theorem
It says when the same side interior angles are supplementary, then the given lines are parallel to each other.
This concept is just converse of the theorem discussed above.
Example
In the above figure, angle y and z are Same Side Interior Angle
\angle y\ = 104 degree
\angle z\ = 76 degree
Adding angle y and z, we get:
\angle y\ + \angle z\ = 180 degree
Since both the angles are supplementary.
Then using corollary of Same Side Interior Angle Theorem, the given lines are parallel to each other.
Same Side Interior Angle Questions
(01) Find angle y, using Same Side Interior Angle Theorem
Angle x and y are same side interior angles
Since the given lines are parallel;
⟹ \angle x\ + \angle y\ = 180 degree
⟹ 78 + \angle y\ = 180 degree
⟹ \angle y\ = 180 – 78
⟹ \angle y\ = 102 degree
(02) Find the value of angle x in the below figure
Angle x and Angle y are same side interior angles
Since both the lines are parallel;
⟹ \angle x\ + \angle y\ = 180 degree
⟹ \angle x\ = 180 – 67
⟹ \angle x\ = 113 degree
(03) Find if the given lines M & N are parallel or not
In the above figure, angle x and angle y are Same Side Interior Angles \angle x\ = 120 degree
\angle y\ = 56 degree
Adding both the angles we get:
\angle x\ + \angle y\ = 120 + 56
\angle x\ + \angle y\ = 176
Using Corollary of Same Side Interior Angles, we can say that:
Since both the angles are not supplementary, the lines M & N are not parallel to each other.
(04) Find value of angle z using Same Side Interior Angle Theorem
Since transversal T is a straight line.
The angle formed will add up to 180 degree
⟹ \angle x\ + \angle y\ = 180
⟹ \angle y\ =180 – 65
⟹ \angle y\ =115
From the image you can observe that angle y & z are same side interior angle
⟹ \angle y\ + \angle z\ = 180
⟹ \angle z\ = 180 – 115
⟹ \angle z\ = 65 degree
Hence, 65 degree is the right answer
(05) In the below image, line P & Q are parallel and line M & N are parallel to each other.
Find the value of angle Z using Same Side Interior Angle Theorem
Since line M & N are parallel to each other.
⟹ \angle x\ = \angle A\ { Corresponding Angles}
⟹ \angle A\ = 72 degree
Since N is a straight line
The sum of angles will be 180 degrees
⟹ \angle A\ + \angle B\ = 180
⟹ \angle B\ = 180 – 72
⟹ \angle B\ = 108 degree
Since line P & Q are parallel to each other, the same side interior angle will be supplementary
⟹ \angle B\ + \angle Z\ = 180
⟹ \angle Z\ = 180 -108
⟹ \angle Z\ = 72 degree
Hence 72 degree is the right answer