Question

explain why the supplement of an acute angle must be obtuse.
if two rays form an acute angle, their opposite rays will form the opposite type of angle, an obtuse angle.
if two angles add up to 180°, and one is an acute angle of greater than 90°, then the other has to be an obtuse angle of less than 90°.
if two angles add up to 180°, then they both have to be less than 180°.
if two angles add up to 180° and one is an acute angle of less than 90°, then the other has to be an obtuse angle of greater than 90°.

Explanation:

To determine the correct explanation for why the supplement of an acute angle must be obtuse, we analyze each option:

Option 1: “If two rays form an acute angle, their opposite rays will form the opposite type of angle, an obtuse angle.”

This focuses on rays forming angles, not on supplementary angles (which are about angle sums, not ray orientation). Eliminate this.

Option 2: “If two angles add up to \( 180^\circ \), and one is an acute angle of greater than \( 90^\circ \), then the other has to be an obtuse angle of less than \( 90^\circ \).”

By definition, an acute angle is less than \( 90^\circ \), not “greater than \( 90^\circ \).” This misdefines acute angles, so it is incorrect.

Option 3: “If two angles add up to \( 180^\circ \), then they both have to be less than \( 180^\circ \).”

While supplementary angles sum to \( 180^\circ \), this statement does not explain why one angle being acute implies the other is obtuse. It is too general and irrelevant to the relationship between acute and obtuse angles. Eliminate this.

Option 4: “If two angles add up to \( 180^\circ \) and one is an acute angle of less than \( 90^\circ \), then the other has to be an obtuse angle of greater than \( 90^\circ \).”

1. Let the acute angle be \( x \), where \( x < 90^\circ \).
2. Let its supplement be \( y \), so \( x + y = 180^\circ \).
3. Solve for \( y \): \( y = 180^\circ - x \).
4. Since \( x < 90^\circ \), subtracting \( x \) from \( 180^\circ \) gives \( y > 180^\circ - 90^\circ = 90^\circ \).
5. Also, \( y < 180^\circ \) (because \( x > 0^\circ \), so \( y = 180^\circ - x < 180^\circ \)).
6. By definition, an angle greater than \( 90^\circ \) and less than \( 180^\circ \) is obtuse. Thus, \( y \) (the supplement) is obtuse.

This option correctly uses the definition of acute angles (\( < 90^\circ \)) and the property of supplementary angles (sum to \( 180^\circ \)) to show the supplement must be obtuse (\( > 90^\circ \) and \( < 180^\circ \)).

Answer:

If two angles add up to \( 180^\circ \) and one is an acute angle of less than \( 90^\circ \), then the other has to be an obtuse angle of greater than \( 90^\circ \). (The fourth option in the list)