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°.
Is this system of axioms consistent? Justify your answer
Solution:
Given, the statements are
(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 formed is equal to 180°.
We have to determine if the system of axioms is consistent or not.
In a pair of intersecting lines, the vertically opposite angles are equal is a theorem
From the figure,
∠AOD = ∠COB
∠AOC = ∠DOB
Therefore, the first statement is false and not an axiom.
From the figure,
AB is a ray standing on a straight line CD.
AB is perpendicular to CD
The adjacent angles are right angles
∠B = ∠ABC + ∠ABD
= 90° + 90°
Total angle = 180°
Therefore, the second statement is true and an axiom.
✦ Try This: In the given diagram light is travelling through different media. It is noted by a scientist that ∠1= ∠3= ∠4 but ∠2 <∠1. State whether the following statement is true or false.
“Medium 3 is the denser than medium 1”
☛ Also Check: NCERT Solutions for Class 9 Maths Chapter 5
NCERT Exemplar Class 9 Maths Exercise 5.4 Problem 4
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°. Is this system of axioms consistent? Justify your answer
Summary:
From the following two statements (i) If two lines intersect each other, then the vertically opposite angles are not equal, is not consistent and not an axiom, (ii) If a ray stands on a line, then the sum of two adjacent angles so formed is equal to 180°, is consistent and an axiom
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