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.

We will find the converse of each statement. Then, we will decide if the converse is true or false. Step 1 — Converse of statement (i) Let's write the converse. The original statement is "If triangle ABC is isosceles, then its base angles are equal." The converse is: If the base angles of triangle ABC are equal, then it is isosceles. This statement is true. If two angles of a triangle are equal, the sides opposite to them are also equal. This means the triangle has two equal sides. Therefore, the triangle is isosceles. > Step 2 — Converse of statement (ii) Let's write the converse. The original statement is "If an integer is odd, then its square is an odd integer." The converse is: If the square of an integer is an odd integer, then the integer is odd. This statement is true. If an integer is even, its square is always even. For example, (even), (even). So, if the square is odd, the integer itself must be odd. > Step 3 — Converse of statement (iii) Let's write the converse. The original statement is "If , then ." The converse is: If , then . This statement is true. If is 1, then will be . So, is correct. > Step 4 — Converse of statement (iv) Let's write the converse. The original statement is "If ABCD is a parallelogram, then AC and BD bisect each other." The converse is: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram. This statement is true. This is a known property of quadrilaterals. If diagonals bisect each other, the quadrilateral is a parallelogram. > Step 5 — Converse of statement (v) Let's write the converse. The original statement is "If , and , are whole numbers, then ." The converse is: If , then , , and are whole numbers. This statement is false. The property is called the associative property of addition. This property holds for many types of numbers. For example, it holds for integers, rational numbers, and real numbers. Consider , , . Here, and are not whole numbers. But the equation is still true. > Step 6 — Converse of statement (vi) Let's write the converse. The original statement is "If and are two odd numbers, then is an even number." The converse is: If is an even number, then and are two odd numbers. This statement is false. The sum of two even numbers is also an even number. For example, let and . Here, is an even number. But and are both even, not odd. > Step 7 — Converse of statement (vii) Let's write the converse. The original statement is "If vertices of a parallelogram lie on a circle, then it is a rectangle." The converse is: If a parallelogram is a rectangle, then its vertices lie on a circle. This statement is true. A rectangle is a special type of parallelogram. All rectangles can be inscribed in a circle. This means all four vertices of a rectangle can lie on a circle. > Answer > (i) Converse: If the base angles of triangle ABC are equal, then it is isosceles. (True) > (ii) Converse: If the square of an integer is an odd integer, then the integer is odd. (True) > (iii) Converse: If , then . (True) > (iv) Converse: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram. (True) > (v) Converse: If , then , , and are whole numbers. (False) > (vi) Converse: If is an even number, then and are two odd numbers. (False) > (vii) Converse: If a parallelogram is a rectangle, then its vertices lie on a circle. (True)

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

We will find the converse of each statement. Then, we will decide if the converse is true or false.

Step 1 — Converse of statement (i)

Let's write the converse. The original statement is "If triangle ABC is isosceles, then its base angles are equal." The converse is: If the base angles of triangle ABC are equal, then it is isosceles.

This statement is true. If two angles of a triangle are equal, the sides opposite to them are also equal. This means the triangle has two equal sides. Therefore, the triangle is isosceles.

$\boxed{\text{Converse: If the base angles of triangle ABC are equal, then it is isosceles. (True)}}$

Step 2 — Converse of statement (ii)

Let's write the converse. The original statement is "If an integer is odd, then its square is an odd integer." The converse is: If the square of an integer is an odd integer, then the integer is odd.

This statement is true. If an integer is even, its square is always even. For example, $2^2 = 4$ (even), $4^2 = 16$ (even). So, if the square is odd, the integer itself must be odd.

$\boxed{\text{Converse: If the square of an integer is an odd integer, then the integer is odd. (True)}}$

Step 3 — Converse of statement (iii)

Let's write the converse. The original statement is "If $x^2 = 1$ , then $x = 1$ ." The converse is: If $x = 1$ , then $x^2 = 1$ .

This statement is true. If $x$ is 1, then $x^2$ will be $1 \times 1$ . $1 \times 1 = 1$ So, $x^2 = 1$ is correct.

$\boxed{\text{Converse: If } x = 1 \text{, then } x^2 = 1\text{. (True)}}$

Step 4 — Converse of statement (iv)

Let's write the converse. The original statement is "If ABCD is a parallelogram, then AC and BD bisect each other." The converse is: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram.

This statement is true. This is a known property of quadrilaterals. If diagonals bisect each other, the quadrilateral is a parallelogram.

$\boxed{\text{Converse: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram. (True)}}$

Step 5 — Converse of statement (v)

Let's write the converse. The original statement is "If $a$ , $b$ and $c$ , are whole numbers, then $a + (b + c) = (a + b) + c$ ." The converse is: If $a + (b + c) = (a + b) + c$ , then $a$ , $b$ , and $c$ are whole numbers.

This statement is false. The property $a + (b + c) = (a + b) + c$ is called the associative property of addition. This property holds for many types of numbers. For example, it holds for integers, rational numbers, and real numbers. Consider $a = \mathbf{0.5}$ , $b = \mathbf{1.5}$ , $c = \mathbf{2}$ . $0.5 + (1.5 + 2) = 0.5 + 3.5 = 4$ $(0.5 + 1.5) + 2 = 2 + 2 = 4$ Here, $a$ and $b$ are not whole numbers. But the equation $a + (b + c) = (a + b) + c$ is still true.

$\boxed{\text{Converse: If } a + (b + c) = (a + b) + c \text{, then } a, b, \text{ and } c \text{ are whole numbers. (False)}}$

Step 6 — Converse of statement (vi)

Let's write the converse. The original statement is "If $x$ and $y$ are two odd numbers, then $x + y$ is an even number." The converse is: If $x + y$ is an even number, then $x$ and $y$ are two odd numbers.

This statement is false. The sum of two even numbers is also an even number. For example, let $x = \mathbf{2}$ and $y = \mathbf{4}$ . $x + y = 2 + 4 = 6$ Here, $x+y$ is an even number. But $x$ and $y$ are both even, not odd.

$\boxed{\text{Converse: If } x + y \text{ is an even number, then } x \text{ and } y \text{ are two odd numbers. (False)}}$

Step 7 — Converse of statement (vii)

Let's write the converse. The original statement is "If vertices of a parallelogram lie on a circle, then it is a rectangle." The converse is: If a parallelogram is a rectangle, then its vertices lie on a circle.

This statement is true. A rectangle is a special type of parallelogram. All rectangles can be inscribed in a circle. This means all four vertices of a rectangle can lie on a circle.

$\boxed{\text{Converse: If a parallelogram is a rectangle, then its vertices lie on a circle. (True)}}$

Answer

(i) Converse: If the base angles of triangle ABC are equal, then it is isosceles. (True) (ii) Converse: If the square of an integer is an odd integer, then the integer is odd. (True) (iii) Converse: If $x = 1$ , then $x^2 = 1$ . (True) (iv) Converse: If the diagonals AC and BD of a quadrilateral ABCD bisect each other, then ABCD is a parallelogram. (True) (v) Converse: If $a + (b + c) = (a + b) + c$ , then $a$ , $b$ , and $c$ are whole numbers. (False) (vi) Converse: If $x + y$ is an even number, then $x$ and $y$ are two odd numbers. (False) (vii) Converse: If a parallelogram is a rectangle, then its vertices lie on a circle. (True)

Question 2

Answer

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EXERCISE A1.5