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.

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Solution

$(i)$ We know that, if two lines intersect each other, then the vertically opposite angles are equal. It is a theorem, so the given statement is false and not an axiom.
Hence, given axiom is not consistent.

$(i i)$ We know that, if a ray stands on a line, then the sum of two adjacent angles so formed is equal to $180^{\circ}$ . It is an axiom. So, given statement is true and an axiom.
Hence, given axiom is not consistent.

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Similar questions

Fill in the blanks so as to make the following statements true:
(i) If one angle of a linear pair is acute, then its other angle will be___.
(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ___.
(iii) If the sum of two adjacent angles is $180^{\circ}$ , then the ___ arms of the two angles are opposite rays.

Q. Fill in the blanks so as to make the following statements true:

(i) If one angle of a linear pair is acute, then its other angle will be ........
(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ..........
(iii) If the sum of two adjacent angles is 180°, then the ........ arms of the two angles are opposite rays.

Q. 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.

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?

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?

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