Appendix 1: Proofs in Mathematics | A1.5

Question 1

EXERCISE A1.5

Write the converses of the following statements. (i) If it is hot in Tokyo, then Sharan sweats a lot.

(ii) If Shalini is hungry, then her stomach grumbles.

(iii) If Jaswant has a scholarship, then she can get a degree.

(iv) If a plant has flowers, then it is alive.

(v) If an animal is a cat, then it has a tail.

Solution

The converse of a statement swaps its hypothesis and conclusion.

Step 1 — Converse for statement (i)

We will find the converse. First, we identify the "if" part. This is the hypothesis. Next, we identify the "then" part. This is the conclusion. To form the converse, we swap these two parts.

$\text{Original statement: If it is hot in Tokyo, then Sharan sweats a lot.}$

$\text{Hypothesis (P): It is hot in Tokyo.}$

$\text{Conclusion (Q): Sharan sweats a lot.}$

$\boxed{\text{Converse: If Sharan sweats a lot, then it is hot in Tokyo.}}$

Step 2 — Converse for statement (ii)

We will find the converse. First, we identify the "if" part. This is the hypothesis. Next, we identify the "then" part. This is the conclusion. To form the converse, we swap these two parts.

$\text{Original statement: If Shalini is hungry, then her stomach grumbles.}$

$\text{Hypothesis (P): Shalini is hungry.}$

$\text{Conclusion (Q): Her stomach grumbles.}$

$\boxed{\text{Converse: If Shalini's stomach grumbles, then she is hungry.}}$

Step 3 — Converse for statement (iii)

We will find the converse. First, we identify the "if" part. This is the hypothesis. Next, we identify the "then" part. This is the conclusion. To form the converse, we swap these two parts.

$\text{Original statement: If Jaswant has a scholarship, then she can get a degree.}$

$\text{Hypothesis (P): Jaswant has a scholarship.}$

$\text{Conclusion (Q): She can get a degree.}$

$\boxed{\text{Converse: If Jaswant can get a degree, then she has a scholarship.}}$

Step 4 — Converse for statement (iv)

We will find the converse. First, we identify the "if" part. This is the hypothesis. Next, we identify the "then" part. This is the conclusion. To form the converse, we swap these two parts.

$\text{Original statement: If a plant has flowers, then it is alive.}$

$\text{Hypothesis (P): A plant has flowers.}$

$\text{Conclusion (Q): It is alive.}$

$\boxed{\text{Converse: If a plant is alive, then it has flowers.}}$

Step 5 — Converse for statement (v)

We will find the converse. First, we identify the "if" part. This is the hypothesis. Next, we identify the "then" part. This is the conclusion. To form the converse, we swap these two parts.

$\text{Original statement: If an animal is a cat, then it has a tail.}$

$\text{Hypothesis (P): An animal is a cat.}$

$\text{Conclusion (Q): It has a tail.}$

$\boxed{\text{Converse: If an animal has a tail, then it is a cat.}}$

Answer

(i) If Sharan sweats a lot, then it is hot in Tokyo. (ii) If Shalini's stomach grumbles, then she is hungry. (iii) If Jaswant can get a degree, then she has a scholarship. (iv) If a plant is alive, then it has flowers. (v) If an animal has a tail, then it is a cat.

More questions in A1.5

EXERCISE A1.5

Write the converses of the following statements. (i) If it is hot in Tokyo, then Sharan sweats a lot.

(ii) If Shalini is hungry, then her stomach grumbles.

(iii) If Jaswant has a scholarship, then she can get a degree.

(iv) If a plant has flowers, then it is alive.

(v) If an animal is a cat, then it has a tail.

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.