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?

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Solution

This system of axioms is not consistent.
(i) When two parallel lines are intersected by a tranversal, then corresponding angles are equal.
Here, $∠ 1$ = $∠ 5$ , $∠ 2$ = $∠ 6$ , $∠ 3$ = $∠ 7$ and $∠ 4$ = $∠ 8$
(ii) The interior angles will always be equal in parallel lines.

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Similar questions

State whether the following axioms are True or False:

If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.

State whether the following axioms are True or False:

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

Q. Read the following two statements which are taken as axioms :
(i) If two lines intersect each other, then the vertically opposite angles are not equal.
(ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to

180^{o}

.
Is this system of axioms consistent?

Which of the following statements are true (I) 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 angles on the same side of the transversal are equal.

Q. 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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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?

This system of axioms is not consistent.(i) When two parallel lines are intersected by a tranversal, then corresponding angles are equal.Here, ∠1 = ∠5, ∠2 = ∠6, ∠3= ∠7 and ∠4= ∠8(ii) The interior angles will always be equal in parallel lines.

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?

This system of axioms is not consistent.
(i) When two parallel lines are intersected by a tranversal, then corresponding angles are equal.
Here, $∠ 1$ = $∠ 5$ , $∠ 2$ = $∠ 6$ , $∠ 3$ = $∠ 7$ and $∠ 4$ = $∠ 8$
(ii) The interior angles will always be equal in parallel lines.