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Chapter 3: Problem 2

Explain why an obtuse angle cannot have a complement.

Short Answer

Expert verified
An obtuse angle is too large to have a complement because it exceeds 90 degrees, preventing any sum of angles being 90.

Step by step solution

01

Understanding Angle Types

First, let's define the types of angles. An obtuse angle is one that measures more than 90 degrees but less than 180 degrees. A complementary angle refers to two angles that sum up to 90 degrees.
02

Definition of Complementary Angles

Complementary angles are two angles whose measures add up to 90 degrees. For example, 30 degrees and 60 degrees are complementary because their sum is 90 degrees.
03

Evaluating the Possibility of Complements for Obtuse Angles

Since an obtuse angle measures more than 90 degrees, adding any positive angle to it would result in a sum greater than 90 degrees. Therefore, it isn't possible for an obtuse angle to have a complementary angle, since no angle can make their sum equal to 90 degrees.
04

Conclusion

Based on the fact that the smallest measure of an obtuse angle is already greater than 90 degrees, it's impossible for it to have a complementary angle, as the definition requires angles to sum to exactly 90 degrees.

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Key Concepts

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

Complementary Angles
Complementary angles are two angles that together sum up to 90 degrees. This can be thought of like two puzzle pieces fitting perfectly together to form a right angle. If you imagine a right angle, which is exactly 90 degrees, complementary angles are the two angles that add up to this right angle.
For example:
  • A 45-degree angle and another 45-degree angle are complementary, because their sum is 90 degrees.
  • Similarly, a 30-degree angle paired with a 60-degree angle are complementary as well.
If you think of these angles in geometry, when you combine complementary angles, they form a corner perfectly. However, if an angle measures more than 90 degrees, it cannot have an angle that adds up to 90 degrees, because it's already over that limit!
Angle Types
There are different types of angles based on their measurements:
  • Acute Angles: These are small angles less than 90 degrees. Imagine a slice of a pizza that's without the rust 鈥� tiny!
  • Right Angles: These are precisely 90 degrees, like the corner of a square that's perfect.
  • Obtuse Angles: These go beyond 90 degrees but are less than 180 degrees. Picture the wide arms of a clock at 10:00.
  • Straight Angles: Exactly 180 degrees, like a flat line.
Understanding these types helps in solving many geometric problems. What's crucial to remember is that an obtuse angle cannot fit into the definition of being complementary due to its measurement exceeding 90 degrees.
Angle Sum Properties
The overall sum of angles in certain shapes can give us a lot of insight. For example, the sum of angles in a triangle is always 180 degrees, much like three straight angles combined. Similarly, when we think of complementary angles,
  • Their property is that together, they must add up to 90 degrees.
  • If you have a complete circle, which is 360 degrees, any two angles on a straight line will sum up to 180 degrees, known as supplementary angles.
Knowing these properties is essential in more complex geometry questions, as it helps piece together how various angles relate. In the context of an obtuse angle, which alone stretches beyond 90 degrees, finding a pair for complementarity isn't possible, since it can't shrink down to make up 90 degrees with its twin angle.

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Most popular questions from this chapter

Writing Math Write a few sentences describing how rulers and protractors are used in geometry.

Angles \(X Y Z\) and \(W Y X\) form a linear pair. If \(m \angle W Y X=56\), what is \(m \angle X Y Z\) ?

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