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.