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Mark the statements that are true.

A. An angle that measures [tex]\(300^{\circ}\)[/tex] is an obtuse angle.
B. An angle that measures [tex]\(\frac{4 \pi}{3}\)[/tex] is a reflex angle.
C. An angle that measures [tex]\(65^{\circ}\)[/tex] is an acute angle.
D. An acute angle measures less than [tex]\(\frac{\pi}{2}\)[/tex].

Sagot :

GinnyAnswer
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
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