Sure, let's analyze each of the statements to determine their validity.
### Statement A:
An angle that measures [tex]\(300^\circ\)[/tex] is an obtuse angle.
To be classified as an obtuse angle, an angle must be greater than [tex]\(90^\circ\)[/tex] but less than [tex]\(180^\circ\)[/tex]. Since [tex]\(300^\circ\)[/tex] is greater than [tex]\(180^\circ\)[/tex], it is not an obtuse angle. Therefore, this statement is false.
Result: False
### Statement B:
An angle that measures [tex]\(\frac{4\pi}{3}\)[/tex] radians is a reflex angle.
A reflex angle is an angle that is greater than [tex]\(180^\circ\)[/tex] (or [tex]\(\pi\)[/tex] radians) but less than [tex]\(360^\circ\)[/tex] (or [tex]\(2\pi\)[/tex] radians). Converting [tex]\(\frac{4\pi}{3}\)[/tex] radians to degrees:
[tex]\[
\frac{4\pi}{3} \times \frac{180^\circ}{\pi} = 240^\circ
\][/tex]
Since [tex]\(240^\circ\)[/tex] is greater than [tex]\(180^\circ\)[/tex], it fits the definition of a reflex angle. Therefore, this statement is true.
Result: True
### Statement C:
An angle that measures [tex]\(65^\circ\)[/tex] is an acute angle.
An acute angle is an angle that measures less than [tex]\(90^\circ\)[/tex]. Since [tex]\(65^\circ\)[/tex] is indeed less than [tex]\(90^\circ\)[/tex], it is an acute angle. Therefore, this statement is true.
Result: True
### Statement D:
An acute angle measures less than [tex]\(\frac{\pi}{2}\)[/tex].
Converting [tex]\(\frac{\pi}{2}\)[/tex] radians to degrees:
[tex]\[
\frac{\pi}{2} \times \frac{180^\circ}{\pi} = 90^\circ
\][/tex]
An acute angle is defined as measuring less than [tex]\(90^\circ\)[/tex]. Since [tex]\(\frac{\pi}{2}\)[/tex] radians is equal to [tex]\(90^\circ\)[/tex], the statement that an acute angle measures less than [tex]\(\frac{\pi}{2}\)[/tex] radians is true.
Result: True
Based on the above analysis:
- Statement A: False
- Statement B: True
- Statement C: True
- Statement D: True
