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Lines Parallel to the Same Line

If a transver...

Question

If a transversal intersects two lines such that a pair of alternate interior angles is equal , then the two lines are parallel.

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Solution

Yes If a transversal intersects two lines such that a pair of alternate interior angles is equal , then the two lines are parallel.

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Q. Fill in the blanks in each of the following to make the statement true:

(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are ...
(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are ....
(iii) Two lines perpendicular to the same line are ... to each other.
(iv) Two lines parallel to the same line are ... to each other.
(v) If a transversal intersects a pair of lines in such away that a pair of alternate angles are equal, then the lines are ...
(vi) If a transversal intersects a pair of lines in such away that the sum of interior angles on the same side of transversal is 180°, then the lines are ...

If two parallel lines are intersected by a transversal, then alternate interior angles are equal.

When a transversal cuts two lines such that pairs of alternate interior angles are equal, then the lines are

Q. Read the following statements which are taken as axioms :
(i) If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
(ii) If a transversal intersect two parallel lines, then alternate interior angles are equal.
Is this system of axioms consistent?

Q. If a transversal intersects two parallel lines then show that the bisectors of any pair of alternate angles are also parallel.

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