if a transversal intersect of two parallel lines
then
show that alternate angle of each
pair of interior angles are equal.
Answers
Step-by-step explanation:
When a transversal intersects two parallel lines, each pair of alternate interior angles is equal. Conversely, if a transversal intersects two lines such that a pair of interior angles is equal, then the two lines are parallel.
Step-by-step explanation:
When a transversal intersects two parallel lines, each pair of alternate interior angles is equal. Conversely, if a transversal intersects two lines such that a pair of interior angles is equal, then the two lines are parallel.
Proof: Refer to the figure above. We have:
∠
1
=
∠
5
(corresponding angles)
∠
3
=
∠
5
(vertically opposite angles)
Thus,
∠
1
=
∠
3
Similarly, we can prove that
∠
2
=
∠
4
. Conversely, suppose that
∠
1
=
∠
3
. We have to prove that the lines are parallel. Since
∠
3
=
∠
5
(vertically opposite angles), we have:
∠
1
=
∠
5
Thus, a pair of corresponding angles is equal, which can only happen if the two lines are parallel.
What about any pair of co-interior angles?
Theorem: If a transversal intersects two parallel lines, then each pair of co-interior angles is supplementary (their sum is 1800). Conversely, if a transversal intersects two lines such that a pair of co-interior angles is supplementary, then the two lines are parallel.
Proof: Refer to the following figure once again: We have:
∠
1
=
∠
5
(corresponding angles)
∠
5
+
∠
4
= 1800 (linear pair)
è
∠
1
+
∠
4
= 1800
Similarly, we can show that
∠
2
+
∠
3
= 1800
The converse part of the proof is left to you as an exercise.
