Read the following statement which is taken as axiom:
$(i)$ If a transversal intersects two parallel lines, then corresponding angles are not necessarily equal.
Is this axiom consistent? Justify your answer.

Read the following statement which is taken as axiom:
Is this axiom consistent? Justify your answer.
$(i i)$ If a transversal intersect two parallel lines, then alternate interior angles are equal.
Is this axiom consistent? Justify your answer.

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Solution

$(i)$ We know that, if a transversal intersects two parallel lines, then each pair of corresponding angles are equal, which is a theorem. So, statement is false and not an axiom.
Hence, given axiom is not consistent.

$(i i)$ We know that, if a transversal intersects two parallel lines, then each pair of alternate interior angles are equal. It is also a theorem. So, Statement is true and a theorem.
Hence, given axiom is not consistent.

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Similar questions

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. 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 ...

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. Read the following statement which is taken as axiom:

(i)

If two lines intersect each other, then the vertically opposite angles are not equal.
Is this axiom consistent? Justify your answer.

Read the following statement which is taken as axiom:

(i i)

If a ray stands on a line, then the sum of two adjacent angles so formed is equal to

180^{\circ}

.
Is this axiom consistent? Justify your answer.

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?

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