Maths-

General

Easy

Question

# Determine if the given conjecture is true or not. Give a counterexample if it is false. Given: ∠𝑃 𝑎𝑛𝑑 ∠𝑄 are complementary. ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary. Conjecture: ∠𝑃 ≅ ∠R

## The correct answer is: As ∠𝑃 and ∠R have equal measures. Hence, the given conjecture i.e. “ ∠𝑃 ≅ ∠R ” is true.

### Hint:

Inductive Reasoning is the process of drawing a general conclusion by observing a pattern based on the observations and this conclusion is called conjecture.

Counterexample: It is an example which shows that the conjecture is false.

Solution

It is given that ∠𝑃 𝑎𝑛𝑑 ∠𝑄 are complementary. ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary.

Draw the figure which shows that ∠𝑃 𝑎𝑛𝑑 ∠𝑄 and also in which ∠𝑄 𝑎𝑛𝑑 ∠𝑅 are complementary.

In this figure

∠𝑃 + ∠𝑄 = 90^{o} …..(1)

and ∠Q + ∠R = 90^{o} …..(2)

Equating (1) and (2)

∠𝑃 + ∠𝑄 = ∠Q + ∠R

∠𝑃 = ∠R

We know that two angles are congruent when they have equal measures. So ∠𝑃 ≅ ∠R.

Final Answer:

As ∠𝑃 and ∠R have equal measures. Hence, the given conjecture i.e. “ ∠𝑃 ≅ ∠R ” is true.

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