Alternate interior angles are the pairs of angles formed when a transversal intersects two parallel or non-parallel lines. They are formed on the inner side of the parallel lines but on the opposite sides of the transversal.
If the alternate interior angles are equal the two lines intersected by the transversal are parallel to each other. On the other hand, alternate interior angles formed when a transversal crosses two non-parallel lines, are found to have no geometric relation.
Alternate Interior Angles
The above figure shows two parallel lines AB and CD intersected by the transversal RS. The two pairs of alternate interior angles formed are:
∠1 and 2
∠3 and ∠4
Thus ∠1 = ∠2 and ∠3 = ∠4
Properties
They are congruent
Angles formed on the same side of the transversal involving two parallel lines are supplementary. These are known as consecutive interior angles
In the case of non-parallel lines, alternate interior angles have no geometric relation with each other
They are supplementary if the transversal intersects two parallel lines at right angles
Alternate Interior Angles Theorem
Alternate Interior Angles Theorem
Prove Alternate Interior Angles Theorem
To prove:
∠3 = ∠6 and ∠4 = ∠5
Proof:
Let ‘PQ’ and ‘RS’ be the two parallel lines intersected by the transversal XY From the properties of parallel lines, we know that if a transversal intersects two parallel lines then the corresponding angles and the vertically opposite ages are equal to each other. Thus, we can write ∠2 = ∠5…… (1) (Corresponding angles) ∠2 = ∠4…… (2) (Vertically opposite angles) From (1) and (2) we get, ∠4 = ∠5 (Alternate interior angles) Similarly, ∠3 = ∠6 Hence Proved
The converse of Alternate Interior Angles Theorem
Converse of Alternate Interior Angles Theorem
Prove the Converse of Alternate Interior Angles Theorem
To prove:
PQ∥ RS
Proof:
Let ∠4 = ∠5 and ∠3 = ∠6 be the two pairs of congruent alternate interior angles Using the above figure, we can write ∠2 = ∠5 (Corresponding angles) Hence PQ is parallel to RS Hence Proved
Let us take some examples to understand the concept better.
As we know the alternate interior angles are congruent Thus, (4x – 15)° = (3x + 12)° 4x – 3x = 12 + 15 x = 27
Given that the two alternate interior angles (4x – 19)° and (3x + 16)° are congruent. Find the value of x and the values of the two alternate interior angles.
Solution:
According to the interior angle theorem, alternate interior angles are equal when the transversal crosses two parallel lines. Thus, (4x – 19)° = (3x + 16)° 4x – 3x = 16 + 19 x = 35° Now, substituting the value of x in both the interior angles expression we get, (4x – 19)° = 4 x 35 – 19 = 121° (3x + 16)° = 3 x 35 + 16 = 121° Thus the values of the two alternate interior angles are 121°
Examples in Real Life
The letter ‘Z’ where the top and the bottom horizontal lines are parallel and the diagonal line is the transversal
Find the value of x and y in the given figure.
Solve the value of x in the given figure.