**Deductive reasoning in geometry worksheet :**

Worksheet given in this section is much useful to the students who would like to practice problems on deductive reasoning in geometry.

**Problem 1 :**

Let p be "the value of x is -5" and let q be "the absolute value of x is 5".

(i) Write p -> q in words.

(ii) Write q -> p in words.

(iii) Decide whether the biconditional statement p <-> q is true.

**Problem 2 :**

Write the following statements using symbols.

(i) ∠A measures 30°

(ii) ∠A does not measure 30°

**Problem 3 :**

Let p be "it is raining" and let q be "the cricket match is cancelled".

(i) Write the contrapositive of p -> q.

(ii) Write the inverse of p -> q.

**Problem 4 :**

State whether the argument is valid.

Michael knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Michael misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday.

**Problem 5 :**

Write some conditional statements that can be made from the following true statements using the Law of Syllogism.

1. If a bird is the fastest bird on land, then it is the largest of all birds.

2. If a bird is the largest of all birds, then it is an ostrich.

3. If a bird is a bee hummingbird, then it is the smallest of all birds.

4. If a bird is the largest of all birds, then it is flightless.

5. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.

**Problem 1 :**

Let p be "the value of x is -5" and let q be "the absolute value of x is 5".

(i) Write p -> q in words.

(ii) Write q -> p in words.

(iii) Decide whether the biconditional statement p <-> q is true.

**Solution : **

(i) If the value of x is -5, then the absolute value of x is 5.

(ii) If the absolute value of x is 5, then the value of x is -5.

(iii) The conditional statement in part (a) is true, but its converse in part (b) is false. So, the biconditional statement p <-> q is false.

**Problem 2 :**

Write the following statements using symbols.

(i) ∠A measures 30°

(ii) ∠A does not measure 30°

**Solution : **

**Statement**

∠A measures 30°

**Symbol**

p

**Negation**

∠A does not measure 30°

**Symbol**

∼ p

**Problem 3 :**

Let p be "it is raining" and let q be "the cricket match is cancelled".

(i) Write the contrapositive of p -> q.

(ii) Write the inverse of p -> q.

**Solution : **

(i) Contrapositive : ∼ q -> ∼ p

If the the cricket match is not cancelled, then it is not raining.

(i) Inverse : ∼ p -> ∼ q

If it is not raining, then the cricket match is not cancelled.

**Problem 4 :**

State whether the argument is valid.

Michael knows that if he misses the practice the day before a game, then he will not be a starting player in the game. Michael misses practice on Tuesday so he concludes that he will not be able to start in the game on Wednesday.

**Solution : **

This logical argument is a valid use of the Law of Detachment. It is given that both a statement (p -> q) and its hypothesis (p) are true. So it is valid for Michael to conclude that the conclusion is true.

**Problem 5 :**

Write some conditional statements that can be made from the following true statements using the Law of Syllogism.

1. If a bird is the fastest bird on land, then it is the largest of all birds.

2. If a bird is the largest of all birds, then it is an ostrich.

3. If a bird is a bee hummingbird, then it is the smallest of all birds.

4. If a bird is the largest of all birds, then it is flightless.

5. If a bird is the smallest bird, then it has a nest the size of a walnut half-shell.

**Solution :**

Here are the conditional statements that use the Law of Syllogism.

a. If a bird is the fastest bird on land, then it is an ostrich. (Use 1 and 2.)

b. If a bird is a bee hummingbird, then it has a nest the size of a walnut half-shell. (Use 3 and 5.)

c. If a bird is the fastest bird on land, then it is flightless. (Use 1 and 4.)

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