Question

select all the counterexamples for the conjecture, \if an angle is not acute, then it is obtuse.\
□ a
diagram of a right angle (perpendicular lines with a right angle symbol)
□ b
diagram of an acute angle (two rays forming a small angle)
□ c
diagram of an obtuse angle (two rays forming a wide angle)
□ d
diagram of a straight angle (a straight line with a ray)
□ e
diagram of a right angle (perpendicular lines with a right angle symbol)

Explanation:

A counterexample to the conjecture "If an angle is not acute, then it is obtuse" is an angle that is not acute (so it's either right, straight, or obtuse, but we need angles that are not acute AND not obtuse).

• Option A: The angle is a right angle (90°). A right angle is not acute (acute < 90°) and not obtuse (obtuse > 90° and < 180°), so it's a counterexample.
• Option B: The angle is acute (less than 90°), so it doesn't satisfy the "not acute" condition, so it's not a counterexample.
• Option C: The angle is obtuse (greater than 90° and less than 180°), so it fits the conjecture (not acute → obtuse), so it's not a counterexample.
• Option D: The angle is a straight angle (180°). A straight angle is not acute and not obtuse (obtuse is less than 180°), so it's a counterexample.
• Option E: The angle is a right angle (90°), similar to Option A, it's not acute and not obtuse, so it's a counterexample.

Answer:

A. The angle is a right angle (90°), not acute and not obtuse.
D. The angle is a straight angle (180°), not acute and not obtuse.
E. The angle is a right angle (90°), not acute and not obtuse.