Using the concept of corresponding angles and vertical angles, let’s derive a relationship between alternate interior angles.
Consider two lines p and q such that p ∥ q
We know that if a transversal intersects any two parallel lines, the pairs of corresponding angles and vertical angles formed are equal.
∠2 and ∠6 form corresponding angles.
Therefore, ∠2 = ∠6
∠2 and ∠3 form vertical angles.
Therefore, ∠2 = ∠3
Hence, we can say that,
∠3 = ∠6
Now ∠3 and ∠6 form a pair of alternate interior angles. Hence, we can conclude that when two parallel lines are intersected by a transversal, the alternate interior angles formed are equal.
Let’s understand this concept better by analyzing the alternate interior angles formed when the intersected lines are parallel and not parallel.
In figure (a), line p and line q are coplanar but not parallel;
That means p ∦ q while line t is the transversal. The pair of alternate interior angles are ∠3 & ∠6 and ∠4 & ∠5. As lines p ∦ q,
∠3 \(\neq\) ∠6 and ∠4 \(\neq\) ∠5.
In figure (b), line p and line q are coplanar and parallel;
That means p ∥ q while line t is the transversal. The pair of alternate interior angles are ∠3 & ∠6 and ∠4 & ∠5. As lines p ∥ q,
∠3 = ∠6 and ∠4 = ∠5.
Hence we can also conclude that when the alternate interior angles are equal, the lines intersected by the transversal will be parallel.
In the given figure, two parallel lines are intersected by a transversal. Find the ∠B and ∠D
Since the 40° angle and ∠D form a pair of alternate interior angles, they are congruent.
So, ∠D = 40°
Similarly, the 140° angle and ∠B form alternate interior angles. Hence, they are equal as well.
So, ∠B = 140°
Example 2: In the figure shown below find the value of x and the value of angle A and B.
Solution: We know that alternate interior angles are congruent.
Therefore, 14x + 2 = 13x + 9
14x – 13x = 9 – 2
x = 7
∠A = 13 x 7 + 9 = 100
∠B = 14 x 7 + 2 = 100
Interior angles are angles formed on the interior of the parallel lines.
Consider the figure below, a transversal intersects two parallel lines. ∠5 and ∠3 and ∠6 and ∠4 are called consecutive interior angles or co-interior angles.
Each pair of consecutive interior angles sums up to 180 degrees. Therefore,
∠3 + ∠5 = 180°
∠4 + ∠6 = 180°
When a transversal crosses a pair of parallel lines, the corresponding angles formed are congruent.
From the figure,
Pairs of corresponding angles: ∠1 and ∠5, ∠3 and ∠7, ∠2 and ∠6, ∠4 and ∠8
Therefore, ∠1 = ∠5, ∠3 = ∠7, ∠2 = ∠ 6, ∠4 = ∠8