Write the converses of the following statements. Also, decide in each case whether the converse is true or false. (i) If triangle ABC is isosceles, then its base angles are equal. (ii) If an integer is odd, then its square is an odd integer. (iii) If , then . (iv) If ABCD is a parallelogram, then AC and BD bisect each other. (v) If , , and are whole numbers, then . (vi) If and are two odd numbers, then is an even number. (vii) If vertices of a parallelogram lie on a circle, then it is a rectangle.

Appendix 1: Proofs in Mathematics | A1.5

Question 2

Write the converses of the following statements. Also, decide in each case whether the converse is true or false.

(i) If triangle ABC is isosceles, then its base angles are equal.

(ii) If an integer is odd, then its square is an odd integer.

(iii) If $x^2 = 1$ , then $x = 1$ .

(iv) If ABCD is a parallelogram, then AC and BD bisect each other.

(v) If $a$ , $b$ , and $c$ are whole numbers, then $a + (b + c) = (a + b) + c$ .

(vi) If $x$ and $y$ are two odd numbers, then $x + y$ is an even number.

(vii) If vertices of a parallelogram lie on a circle, then it is a rectangle.

Solution

Understand the Question

For a conditional statement of the form "If $p$ , then $q$ ", its converse is formed by interchanging the hypothesis and conclusion: "If $q$ , then $p$ ".
A statement and its converse are independent: a converse can be true or false regardless of whether the original statement is true.
To prove a converse is true, provide a valid mathematical reasoning or property; to prove it is false, a single counterexample is sufficient.

(i) If triangle ABC is isosceles, then its base angles are equal.

Step 1 · Write the Converse and Check Truth Value

Converse: If the base angles of triangle ABC are equal, then it is isosceles.

In any triangle, if two angles are equal, the sides opposite to those angles are also equal. Therefore, the triangle has two equal sides, making it an isosceles triangle.

Hence, the converse is true.

Answer

(i) Converse: If the base angles of triangle ABC are equal, then it is isosceles. (True)

(ii) If an integer is odd, then its square is an odd integer.

Step 1 · Write the Converse and Check Truth Value

Converse: If the square of an integer is an odd integer, then the integer is odd.

The square of an even integer is always even ( $2^2 = 4$ , $4^2 = 16$ ). Therefore, if the square of an integer is odd, the integer itself must be odd.

Hence, the converse is true.

Answer

(ii) Converse: If the square of an integer is an odd integer, then the integer is odd. (True)

(iii) If $x^2 = 1$ , then $x = 1$ .

Step 1 · Write the Converse and Check Truth Value

Converse: If $x = 1$ , then $x^2 = 1$ .

If $x = 1$ : $x^2 = 1^2 = 1$

Hence, the converse is true.

Answer

(iii) Converse: If $x = 1$ , then $x^2 = 1$ . (True)

(iv) If ABCD is a parallelogram, then AC and BD bisect each other.

Step 1 · Write the Converse and Check Truth Value

Converse: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram.

By the geometric property of quadrilaterals, if the diagonals of a quadrilateral bisect each other, the quadrilateral is always a parallelogram.

Hence, the converse is true.

Answer

(iv) Converse: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram. (True)

(v) If $a$ , $b$ , and $c$ are whole numbers, then $a + (b + c) = (a + b) + c$ .

Step 1 · Write the Converse and Check Truth Value

Converse: If $a + (b + c) = (a + b) + c$ , then $a$ , $b$ , and $c$ are whole numbers.

The associative property of addition holds for all real numbers, rational numbers, and integers, not just whole numbers.

Counterexample: Let $a = 0.5$ , $b = 1.5$ , and $c = 2$ :

\begin{aligned} 0.5 + (1.5 + 2) &= 0.5 + 3.5 = 4 \\[0.6em] (0.5 + 1.5) + 2 &= 2 + 2 = 4 \end{aligned}

Here, $a$ and $b$ are not whole numbers, but the equality holds.

Hence, the converse is false.

Answer

(v) Converse: If $a + (b + c) = (a + b) + c$ , then $a$ , $b$ , and $c$ are whole numbers. (False)

(vi) If $x$ and $y$ are two odd numbers, then $x + y$ is an even number.

Step 1 · Write the Converse and Check Truth Value

Converse: If $x + y$ is an even number, then $x$ and $y$ are two odd numbers.

Counterexample: Let $x = 2$ and $y = 4$ (both even numbers): $x + y = 2 + 4 = 6$ The sum $6$ is even, but $x$ and $y$ are not odd.

Hence, the converse is false.

Answer

(vi) Converse: If $x + y$ is an even number, then $x$ and $y$ are two odd numbers. (False)

(vii) If vertices of a parallelogram lie on a circle, then it is a rectangle.

Step 1 · Write the Converse and Check Truth Value

Converse: If a parallelogram is a rectangle, then its vertices lie on a circle.

In a rectangle, opposite angles sum to $180^\circ$ ( $90^\circ + 90^\circ = 180^\circ$ ). Therefore, every rectangle is a cyclic quadrilateral and its vertices always lie on a circle.

Hence, the converse is true.

Answer

(vii) Converse: If a parallelogram is a rectangle, then its vertices lie on a circle. (True)

Common Mistakes

Assuming Truth Values Match: Assuming that if a statement is true, its converse must also be true. For example, in (v) and (vi), the original statements are true, but their converses are false.
Converse vs Negation: Writing the negative/inverse ("If not $p$ , then not $q$ ") instead of simply swapping the hypothesis and conclusion ("If $q$ , then $p$ ").
Overlooking Counterexamples: For proving a converse false, students often try to write long algebraic arguments instead of providing a simple counterexample.

Question 2

More questions in A1.5

EXERCISE A1.5