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Read the following statements which are...

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 ? Justify your answer.

Text Solution

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To determine whether the given system of axioms is consistent, we need to analyze each statement and see if they can coexist without contradiction. ### Step-by-Step Solution: 1. **Understanding the Statements:** - **Statement (i):** If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal. - **Statement (ii):** If a transversal intersects two parallel lines, then alternate interior angles are equal. ...
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Theorem 6.2 : If a transversal intersects two parallel lines, then each pair of alternate interior angles is equal.

Theorem 6.4 : If a transversal intersects two parallel lines, then each pair of interior angles on the same side of the transversal is supplementary.

Corresponding angles between parallel lines are equal

Which of the following statements are true (T) and which are false (F)? Give reasons. (i)If two lines are intersected by a transversal, then corresponding angles are equal. (ii)If two parallel lines are intersected by a transversal, then alternate interior angles are equal. (iii)Two lines perpendicular to the same line are perpendicular to each other. (iv)Two lines parallel to the same line are parallel to each other. (v)If two parallel lines are intersected by a transversal, then the interior angle on the same side of the transversal are equal.

Theorem 6.3 : If a transversal intersects two lines such that a pair of alternate interior angles is equal, then the two lines are parallel.

A transversal intersects two parallel lines. Prove that the bisectors of any pair of corresponding angles so formed are parallel.

If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel.

If two lines intersect prove that the vertically opposite angles are equal

Theorem 6.5 : If a transversal intersects two lines such that a pair of interior angles on the same side of the transversal is supplementary, then the two lines are parallel.

Alternate interior angles between parallel lines are equal