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If a transversal intersects two parallel lines, then prove that each pair of alternate angles are equal.

Last updated date: 09th Sep 2024
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Hint: Starting with drawing a diagram using all the information given in the question. Assume two parallel lines as ‘m’ and ‘n’. And name all the angles formed by the intersection of transverse with parallel lines. Use the property that says that the corresponding angles and opposite angles are equal for a pair of parallel lines. Combine the two hence formed equations to get the required relationship.

Here in this problem, we are given two parallel lines and a transversal intersects it. Now within this situation, we need to prove that each pair of alternate angles is equal.
Before starting with the solution to the above problem, we must draw a figure using all the information given to us.

Alternate angles are the two angles, formed when a line crosses two other lines, that lie on opposite sides of the transversal line and on opposite relative sides of the other lines.
So according to the above figure, we assumed that the two parallel lines be ‘m’ and ‘n’ with one transverse cutting both of them at two points and forming eight angles named as $\angle 1,\angle 2,\angle 3,\angle 4,\angle 5,\angle 6,\angle 7{\text{ and }}\angle 8$ .
Here we need to prove that the pair alternate angles, i.e. $\angle 3 = \angle 5{\text{ and }}\angle 4 = \angle 6$
As we know that the corresponding angles between two parallel lines are equal in measure. A corresponding angle is the angles that occupy the same relative position at each intersection where a straight line crosses two others.
i.e. $\angle 2 = \angle 6,\angle 1 = \angle 5,\angle 3 = \angle 7{\text{ and }}\angle 4 = \angle 8$ …..….(i)
Also, the opposite angles formed in the intersection of two lines are equal in measure,
i.e. $\angle 1 = \angle 3,\angle 2 = \angle 4,\angle 6 = \angle 8{\text{ and }}\angle 5 = \angle 7$ ……..(ii)
From relation (i) and (ii), we can write two pairs as $\angle 2 = \angle 6{\text{ and }}\angle 2 = \angle 4$
Thus, by combining these two equations we can say $\angle 6 = \angle 4$
Similarly, if we take two pairs from (i) and (ii) as $\angle 1 = \angle 5$ and $\angle 1 = \angle 3$
From these above equations, we can conclude that $\angle 3 = \angle 5$
Therefore, we get the required relation $\angle 3 = \angle 5{\text{ and }}\angle 4 = \angle 6$
Hence, we proved that the alternate angles of two parallel lines intersected by a transverse are equal in measure.

Note: Remember that the original problem was to prove that the alternate angles are equal for two parallel lines and an intersecting transverse. We here used our assumption of two lines ‘m’ and ‘n’ and a transverse using a figure. Naming the angles formed on the intersection of these lines helped us to form the relations and equations. Questions like this method are always useful. An alternative approach can be to use the theorem that says that the sum of interior angles on the same side is supplementary