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.

Answer 1

(i) The given statement is: p: “If all the angles of a triangle are equal, then the triangle is an obtuse angled triangle.” Break the statement into parts.

q: All the angles of a triangle are equal.

r: The triangle is an obtuse angled triangle.

To show the given statement p is not true, it has to be proved that if q is true, then r is not true.

For this, consider an example as follows:

Since the sum of all angles of a triangle is 180° , therefore if all the angles are equal, then the measure of each angle will be 60° and it is known that it is not an obtuse angle.

Similarly, if there is an equilateral triangle, then it is seen that each angle is measured equal but it is not an obtuse angle.

Thus, by above examples, it is clear that the statement p is not true.

(ii) The given statement is: q: “The equation x 2 −1=0 does not have a root lying between 0 and 2.” To prove the given statement is not true, consider an example as follows:

x 2 −1=0 x 2 =1 x=±1

Since the root 1 lies between 0 and 2, thus the given statement q is false.