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Chapter 8: Problem 52

Can two complementary angles both be obtuse? Why or why not?

Short Answer

Expert verified
No, two complementary angles cannot both be obtuse because their measures would exceed 90 degrees.

Step by step solution

01

- Understand Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. Mathematically, if two angles \(a\) and \(b\) are complementary, then \(a + b = 90^\text{掳}\).
02

- Define Obtuse Angles

An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees.
03

- Analyze the Possibility

Considering that a complementary angle must add up to 90 degrees, check if an obtuse angle can be part of this combination. Since an obtuse angle is greater than 90 degrees by definition, it's impossible for their sum to be 90 degrees. Therefore, two obtuse angles cannot add up to be complementary.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Obtuse Angles
An obtuse angle is one that measures between 90 and 180 degrees. This means it is larger than a right angle (90 degrees) but smaller than a straight angle (180 degrees). Because of this property, an obtuse angle appears more 'open' or 'spread out' compared to a right angle or acute angle (which is less than 90 degrees).
To visualize an obtuse angle, think of the hands of a clock at 10:10. The angle between the hour and minute hands is obtuse.
Understanding this is crucial as it directly relates to why two obtuse angles can't be complementary. Since an obtuse angle is always larger than 90 degrees, adding another angle to it will always result in a sum greater than 90 degrees.
Angle Measurements
Angles are measured in degrees, a unit that ranges from 0 to 360. Here are some important types of angles commonly encountered:
  • Acute Angles: measure less than 90 degrees.
  • Right Angles: exactly 90 degrees.
  • Obtuse Angles: greater than 90 degrees but less than 180 degrees.
  • Straight Angles: exactly 180 degrees.
  • Reflex Angles: greater than 180 degrees.
Knowing the measurements allows us to categorize angles and understand how they interact. For instance, complementary angles always sum to 90 degrees, so they must each be less than or equal to 90 degrees.
Sum of Angles
When discussing angle sums, it's essential to understand that the combination of certain types of angles leads to specific total measurements. With complementary angles, their sum is always 90 degrees. This is a fixed rule.
When two angles are complementary, their measurements add up to form a right angle. If one angle is known, you can always find the other by subtracting the known angle from 90 degrees.
This property makes it clear why two obtuse angles can never be complementary. Each obtuse angle is already more than 90 degrees, and their sum will always exceed 180 degrees, making it impossible for any two obtuse angles to sum to 90 degrees.

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