Which Of The Following Angle Measurements Is Acute?A. 20° B. 90° C. 110° D. 180°
What are Acute Angles?
In mathematics, an acute angle is an angle whose measure is greater than 0° and less than 90°. Acute angles are one of the three types of angles, the other two being right angles (90°) and obtuse angles (greater than 90° and less than 180°). Understanding acute angles is crucial in various mathematical concepts, including geometry, trigonometry, and calculus.
Defining Acute Angles
An acute angle is defined as an angle whose measure is greater than 0° and less than 90°. This means that any angle that is less than 90° is considered an acute angle. For example, 30°, 45°, and 60° are all acute angles.
Properties of Acute Angles
Acute angles have several properties that distinguish them from other types of angles. Some of the key properties of acute angles include:
- Measure: The measure of an acute angle is greater than 0° and less than 90°.
- Complement: The complement of an acute angle is a right angle (90°).
- Supplement: The supplement of an acute angle is an obtuse angle (greater than 90° and less than 180°).
- Addition: The sum of two acute angles is always greater than 90°.
Examples of Acute Angles
Here are some examples of acute angles:
- 20°
- 30°
- 45°
- 60°
- 80°
Which of the Following Angle Measurements is Acute?
Now, let's look at the options provided in the question:
A. 20° B. 90° C. 110° D. 180°
Based on the definition of an acute angle, we can determine that:
- Option A (20°) is an acute angle.
- Option B (90°) is a right angle, not an acute angle.
- Option C (110°) is an obtuse angle, not an acute angle.
- Option D (180°) is a straight angle, not an acute angle.
Therefore, the correct answer is:
A. 20°
Conclusion
In conclusion, an acute angle is an angle whose measure is greater than 0° and less than 90°. Understanding acute angles is crucial in various mathematical concepts, including geometry, trigonometry, and calculus. By recognizing the properties and examples of acute angles, we can better appreciate the importance of these angles in mathematics.
Key Takeaways
- An acute angle is an angle whose measure is greater than 0° and less than 90°.
- The complement of an acute angle is a right angle (90°).
- The supplement of an acute angle is an obtuse angle (greater than 90° and less than 180°).
- The sum of two acute angles is always greater than 90°.
Frequently Asked Questions
- What is an acute angle?
- What is the measure of an acute angle?
- What is the complement of an acute angle?
- What is the supplement of an acute angle?
References
- "Geometry" by Michael Artin
- "Trigonometry" by I.M. Gelfand
- "Calculus" by Michael Spivak
Acute Angle Q&A =====================
Frequently Asked Questions
Q: What is an acute angle?
A: An acute angle is an angle whose measure is greater than 0° and less than 90°.
Q: What is the measure of an acute angle?
A: The measure of an acute angle is any value greater than 0° and less than 90°. For example, 30°, 45°, and 60° are all acute angles.
Q: What is the complement of an acute angle?
A: The complement of an acute angle is a right angle (90°). For example, if an acute angle measures 30°, its complement is 60° (90° - 30° = 60°).
Q: What is the supplement of an acute angle?
A: The supplement of an acute angle is an obtuse angle (greater than 90° and less than 180°). For example, if an acute angle measures 30°, its supplement is 150° (180° - 30° = 150°).
Q: Can an acute angle be a right angle?
A: No, an acute angle cannot be a right angle. A right angle is an angle whose measure is exactly 90°, which is not within the range of an acute angle.
Q: Can an acute angle be an obtuse angle?
A: No, an acute angle cannot be an obtuse angle. An obtuse angle is an angle whose measure is greater than 90° and less than 180°, which is not within the range of an acute angle.
Q: Can an acute angle be a straight angle?
A: No, an acute angle cannot be a straight angle. A straight angle is an angle whose measure is exactly 180°, which is not within the range of an acute angle.
Q: Can two acute angles add up to 90°?
A: No, two acute angles cannot add up to 90°. The sum of two acute angles is always greater than 90°.
Q: Can two acute angles add up to 180°?
A: No, two acute angles cannot add up to 180°. The sum of two acute angles is always greater than 90°, and the maximum sum is 180°, which is achieved when the two angles are equal (90° each).
Q: What is the relationship between acute angles and trigonometry?
A: Acute angles are used extensively in trigonometry, particularly in the study of triangles and the relationships between their sides and angles.
Q: What is the relationship between acute angles and calculus?
A: Acute angles are used in calculus, particularly in the study of limits and derivatives.
Q: Can acute angles be used in real-world applications?
A: Yes, acute angles are used in various real-world applications, including architecture, engineering, and physics.
Q: What are some examples of acute angles in real-world applications?
A: Some examples of acute angles in real-world applications include:
- The angle between two adjacent sides of a building
- The angle between two adjacent sides of a bridge
- The angle between two adjacent sides of a machine part
- The angle between two adjacent sides of a solar panel
Conclusion
In conclusion, acute angles are an essential concept in mathematics, particularly in geometry, trigonometry, and calculus. Understanding acute angles is crucial in various real-world applications, including architecture, engineering, and physics. By recognizing the properties and examples of acute angles, we can better appreciate the importance of these angles in mathematics.
Key Takeaways
- An acute angle is an angle whose measure is greater than 0° and less than 90°.
- The complement of an acute angle is a right angle (90°).
- The supplement of an acute angle is an obtuse angle (greater than 90° and less than 180°).
- The sum of two acute angles is always greater than 90°.
References
- "Geometry" by Michael Artin
- "Trigonometry" by I.M. Gelfand
- "Calculus" by Michael Spivak