Question

if two sides of adjacent acute angles are perpendicular, then the angles are supplementary.
○ false
○ true

Explanation:

Step1: Recall angle relationships

Adjacent angles with perpendicular sides: If two sides of adjacent angles are perpendicular, the angles form a right angle between them? Wait, no—wait, adjacent acute angles with perpendicular sides. Let's think about the definition of supplementary angles (sum to \(180^\circ\)) and complementary (sum to \(90^\circ\)). Acute angles are less than \(90^\circ\). If two sides are perpendicular, the angle between those sides is \(90^\circ\), but adjacent acute angles: let's visualize. Suppose we have two adjacent angles, each acute, with their outer sides perpendicular. Wait, no—if two sides (of adjacent angles) are perpendicular, the angles formed would be such that the sum? Wait, adjacent angles with perpendicular sides: if two sides are perpendicular, the angle between them is \(90^\circ\), but adjacent acute angles. Wait, supplementary angles sum to \(180^\circ\), but acute angles are less than \(90^\circ\), so two acute angles can sum to at most less than \(180^\circ\) (but actually, two acute angles sum to less than \(180^\circ\), but if their sides are perpendicular, let's think of a right angle. Wait, no—let's take an example. Suppose we have two adjacent acute angles, say \( \angle A \) and \( \angle B \), with one side of \( \angle A \) and one side of \( \angle B \) perpendicular. So the angle between those sides is \(90^\circ\), but the adjacent angles: if they are adjacent, their non-common sides form a right angle? Wait, no—adjacent angles share a common side. If two sides (one from each angle, not the common side) are perpendicular, then the sum of the angles? Wait, acute angles are less than \(90^\circ\), so two acute angles sum to less than \(180^\circ\), but supplementary angles sum to \(180^\circ\). So if two angles are acute (less than \(90^\circ\)), their sum is less than \(180^\circ\), so they can't be supplementary. Wait, but the statement says "if two sides of adjacent acute angles are perpendicular, then the angles are supplementary". Let's check the logic. Adjacent angles: share a common side, and their non-common sides form a linear pair? No, adjacent angles just share a common vertex and side. If two sides (non-common) are perpendicular, the angle between them is \(90^\circ\), but the sum of the two acute angles: let's say angle 1 is \( \alpha \), angle 2 is \( \beta \), both acute (\( \alpha < 90^\circ \), \( \beta < 90^\circ \)). If their non-common sides are perpendicular, then \( \alpha + \beta = 90^\circ \)? Wait, no—wait, if two sides are perpendicular, the angle between them is \(90^\circ\), but adjacent acute angles: maybe the sum is \(90^\circ\) (complementary) or \(180^\circ\) (supplementary). But since they are acute, sum can't be \(180^\circ\) (because each is less than \(90^\circ\), so sum less than \(180^\circ\), but actually, two acute angles sum to less than \(180^\circ\), but to be supplementary, sum must be \(180^\circ\). So the statement is false. Wait, but let's re-examine. Wait, adjacent angles with perpendicular sides: if two sides are perpendicular, the angle between them is \(90^\circ\), but adjacent angles: maybe the sum is \(90^\circ\) (complementary) or \(180^\circ\) (supplementary). But acute angles are less than \(90^\circ\), so two acute angles can't sum to \(180^\circ\) (since each is less than \(90^\circ\), sum is less than \(180^\circ\)). Therefore, the statement is false.

Step2: Evaluate the options

The statement claims that if two sides of adjacent acute angles are perpendicular, the angles are supplementary. But supplementary angles sum to \(180^\circ\), and acute angles are less than \(90^\circ\), so their sum is less than \(180^\circ\), so the statement is false. So the correct option is "False".

Answer:

False (the option: False)