# By giving a counter example, show that the following statements are not true.

Question:

By giving a counter example, show that the following statements are not true.

(i) p: If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.

(ii) $q$ : The equation $x^{2}-1=0$ does not have a root lying between 0 and 2 .

Solution:

(i) The given statement is of the form “if q then r”.

q: All the angles of a triangle are equal.

r: The triangle is an obtuse-angled triangle.

The given statement p has to be proved false. For this purpose, it has to be proved that if q, then ∼r.

To show this, angles of a triangle are required such that none of them is an obtuse angle.

It is known that the sum of all angles of a triangle is 180°. Therefore, if all the three angles are equal, then each of them is of measure 60°, which is not an obtuse angle.

In an equilateral triangle, the measure of all angles is equal. However, the triangle is not an obtuse-angled triangle.

Thus, it can be concluded that the given statement p is false.

(ii) The given statement is as follows.

$q$ : The equation $x^{2}-1=0$ does not have a root lying between 0 and 2 .

This statement has to be proved false. To show this, a counter example is required.

Consider $x^{2}-1=0$

$x^{2}=1$

$x=\pm 1$

One root of the equation $x^{2}-1=0$. i.e the root $x=1$ lies between 0 and 2

Thus, the given statement is false.