Question

question 8
your answer is incorrect.
given: ∠1 and ∠2 are complementary.
what conclusion(s) follow from the given information?
i. ∠1 and ∠2 are adjacent angles.
ii. ∠1 is a right angle and ∠2 is a right angle.

Explanation:

Step1: Recall the definition of complementary angles

Complementary angles are two angles whose sum is \(90^\circ\) (a right angle). If \(\angle1\) and \(\angle2\) are complementary, then \(\angle1+\angle2 = 90^\circ\). But this does not mean they are adjacent (adjacent angles share a common side and vertex) or that each is a right angle (a right angle is \(90^\circ\), if both were right angles their sum would be \(180^\circ\), not \(90^\circ\) for complementary).

Step2: Analyze Conclusion I

Adjacent angles are defined by their position (sharing a common side and vertex), while complementary angles are defined by their sum. Just because two angles are complementary does not mean they are adjacent. For example, two angles in different parts of a diagram can add up to \(90^\circ\) without sharing a side or vertex. So Conclusion I is incorrect.

Step3: Analyze Conclusion II

If \(\angle1\) and \(\angle2\) are complementary, \(\angle1+\angle2=90^\circ\). A right angle is \(90^\circ\), so it's impossible for both \(\angle1\) and \(\angle2\) to be right angles (since \(90^\circ + 90^\circ=180^\circ
eq90^\circ\)). So Conclusion II is incorrect. But the question says "Your answer is INCORRECT" and we need to find which conclusion is wrong? Wait, actually, we need to see which conclusion does not follow. Wait, the given is \(\angle1\) and \(\angle2\) are complementary. Let's re - evaluate:

Wait, maybe I misread. The question is to find which conclusion(s) follow? No, the user says "Your answer is INCORRECT" so we need to find the incorrect conclusion? Wait, the problem is: Given \(\angle1\) and \(\angle2\) are complementary. Which conclusion(s) follow? I. \(\angle1\) and \(\angle2\) are adjacent angles. II. \(\angle1\) is a right angle and \(\angle2\) is a right angle.

For I: Complementary angles do not have to be adjacent. Adjacent is a positional relationship, complementary is an angular sum relationship. So I does not follow.

For II: If \(\angle1\) and \(\angle2\) are complementary, \(\angle1+\angle2 = 90^\circ\). A right angle is \(90^\circ\), so \(\angle1\) and \(\angle2\) can't both be right angles (as \(90 + 90=180
eq90\)). So II does not follow. But the question says "Your answer is INCORRECT" so maybe the task is to find which conclusion is wrong, and both I and II are wrong, but let's check the options. Wait, the problem is to identify the incorrect conclusion. Let's confirm:

• Conclusion I: Complementary \(

Rightarrow\) Adjacent. So I is incorrect.

• Conclusion II: Complementary \(

Rightarrow\) both right angles (since sum is \(90\), not \(180\) for two right angles). So II is incorrect. But the question is probably asking which conclusion is wrong, and both are wrong, but let's see the original problem's "Your answer is INCORRECT" implies that maybe we need to find that both I and II are incorrect, but let's focus on the conclusion.

Wait, the key is: Complementary angles sum to \(90^\circ\). Adjacent angles are about position, not sum. So I is wrong. Two right angles sum to \(180^\circ\), not \(90^\circ\), so II is wrong. But the question is to find the incorrect conclusion, and both are incorrect, but let's answer based on the analysis.

Answer:

Both I and II are incorrect. But if we have to choose the most incorrect or the one that doesn't follow, let's see:

Conclusion II: If \(\angle1\) and \(\angle2\) are complementary, \(\angle1+\angle2 = 90^\circ\). A right angle is \(90^\circ\), so \(\angle1\) and \(\angle2\) cannot both be right angles (as \(90^\circ+90^\circ = 180^\circ
eq90^\circ\)). So Conclusion II is incorrect.

Conclusion I: Complementary angles do not have to be adjacent. So Conclusion I is incorrect.

But since the problem says "Your answer is INCORRECT", and we need to find the incorrect conclusion, the answer is that both I and II are incorrect, but if we take the options as I and II, the incorrect conclusions are I. \(\angle1\) and \(\angle2\) are adjacent angles and II. \(\angle1\) is a right angle and \(\angle2\) is a right angle.