An obtuse angle is an angle that measures greater than 90 degrees but less than 180 degrees. This type of angle is larger than a right angle, which measures exactly 90 degrees, and smaller than a straight angle, which measures exactly 180 degrees. Understanding obtuse angles is essential in various applications such as geometry, trigonometry, and real-world contexts like architecture and design.
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Obtuse angles can be found in various geometric shapes such as triangles, where they can be one of the interior angles.
The sum of the angles in any triangle, including those with an obtuse angle, is always 180 degrees.
Obtuse angles are used in trigonometric functions to determine values for sine, cosine, and tangent.
In a coordinate plane, an obtuse angle can be represented by two rays that diverge to form the larger angle from one ray to another.
When measuring angles in radians, an obtuse angle corresponds to a measurement between $$\frac{\pi}{2}$$ and $$\pi$$.
Review Questions
How do you determine if an angle is classified as obtuse based on its measurement?
An angle is classified as obtuse if its measurement is greater than 90 degrees and less than 180 degrees. You can check this by measuring the angle with a protractor or by comparing it to known angles. If it falls within this range, it qualifies as an obtuse angle.
What is the significance of the sum of angles in a triangle containing an obtuse angle?
In any triangle containing an obtuse angle, the sum of all interior angles must always equal 180 degrees. This means that the other two angles must be acute angles since they must add up to less than 90 degrees to complement the obtuse angle. This relationship helps in solving problems related to triangle properties and classifications.
Evaluate how the presence of an obtuse angle in a triangle influences its overall classification and properties.
The presence of an obtuse angle in a triangle classifies it as an obtuse triangle. This specific classification affects its properties, such as the range of possible side lengths and relationships between sides and angles. For instance, in an obtuse triangle, the side opposite the obtuse angle is always the longest side. This fundamental understanding plays a crucial role in geometry and helps in solving complex problems related to triangles.
Related terms
Acute Angle: An acute angle is an angle that measures less than 90 degrees.
Right Angle: A right angle is an angle that measures exactly 90 degrees.
Straight Angle: A straight angle is an angle that measures exactly 180 degrees.