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Chapter 1: Problem 76

A right triangle cannot contain an obtuse angle.

Short Answer

Expert verified
A right triangle cannot have an obtuse angle, as the remaining angles must sum to 90 degrees.

Step by step solution

01

Understand the Types of Angles

A right triangle contains one right angle, which is exactly 90 degrees. An obtuse angle, on the other hand, is any angle greater than 90 degrees but less than 180 degrees.
02

Review Triangle Angle Sum Property

In any triangle, the sum of all interior angles is 180 degrees. This means that in a right triangle, the remaining two angles aside from the right angle must sum up to 90 degrees.
03

Check the Compatibility of Obtuse Angle in Right Triangle

If a triangle has an angle of 90 degrees (right angle), the other two angles must sum up to 90 degrees, as the total is 180 degrees. Therefore, it's impossible for any of these other angles to be obtuse, as an obtuse angle is greater than 90 degrees.

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Key Concepts

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

Understanding Types of Angles
Angles are an important part of geometry, and they vary based on their degree measures. In geometry, there are several types of angles:
  • Acute Angle: An angle that is less than 90 degrees.
  • Right Angle: An angle that is exactly 90 degrees.
  • Obtuse Angle: An angle that is greater than 90 degrees but less than 180 degrees.
A right triangle specifically features one right angle, defining its unique property. This means that the remaining angles in a right triangle must compensate for the right angle's presence to maintain the overall triangle angle sum property. An obtuse angle, meanwhile, is too large to coexist with a right angle in a right triangle, as its measure exceeds that of a right angle.
Triangle Angle Sum Property
One of the foundational properties of triangles is the triangle angle sum property. This property states that the sum of all interior angles within any triangle equals 180 degrees.
In the context of a right triangle, the right angle itself consumes 90 degrees. Therefore, the other two angles together must add up to another 90 degrees to complete the full 180-degree requirement.
  • If one angle is exactly 90 degrees (right angle), the other two angles combined must also be 90 degrees.
  • This property helps in determining unknown angles when at least one angle is known.
Using this property, we know that if a triangle has a right angle, it can only possess acute angles (less than 90 degrees) along with it. This ensures the sum can only reach, but never exceed, the 180-degree limit.
Why Right Triangles Can't Have Obtuse Angles
It's impossible for a right triangle to possess an obtuse angle due to the limitations of the triangle angle sum property. Let鈥檚 dive a bit deeper:
  • With one angle already consuming 90 degrees (the right angle), the other two angles must sum to only 90 degrees to meet the 180-degree requirement.
  • An obtuse angle is any angle greater than 90 degrees, making it incompatible within a right triangle.
Therefore, if a triangle includes a right angle, it automatically means the remaining angles must be smaller than 90 degrees.
This is why one of the essential rules of geometry tells us that a right triangle cannot have an obtuse angle at all. Each angle in a right triangle must perfectly fit within the 180-degree framework without exceeding it.

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Most popular questions from this chapter

\(C=80^{\circ}\), find \(F\).

Calculate \(\frac{\cos \left(33^{\circ} 38^{\prime} 1^{\prime \prime}\right)}{\csc \left(50^{\circ} 33^{\prime} 2^{*}\right)}-\cot \left(36^{\circ} 58^{\prime} 6^{\prime \prime}\right)\)

Use a calculator to find \(\cos ^{-1}\left(\cos 17^{\circ}\right)\).

\(\cot 55^{\circ}\)

\(\csc \left(28^{\circ} 25^{\prime} 35^{\prime \prime}\right)\)
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