An obtuse triangle is a type of triangle that contains one angle measuring greater than 90 degrees. This characteristic differentiates it from other triangles, such as acute triangles, which have all angles less than 90 degrees, and right triangles, which contain one angle exactly equal to 90 degrees. The presence of an obtuse angle fundamentally affects the triangle's properties and relationships among its sides.
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In an obtuse triangle, the side opposite the obtuse angle is the longest side due to the properties of angles and sides in triangles.
The sum of all interior angles in any triangle, including obtuse triangles, is always equal to 180 degrees.
An obtuse triangle cannot be equilateral because an equilateral triangle has all angles equal to 60 degrees.
Obtuse triangles can be classified as scalene or isosceles, depending on whether they have sides of different lengths or at least two sides of equal length, respectively.
If one angle in a triangle is obtuse, it automatically means that the other two angles must be acute since their total must still equal 180 degrees.
Review Questions
Compare and contrast obtuse triangles with acute and right triangles, focusing on their defining characteristics.
Obtuse triangles are defined by having one angle greater than 90 degrees, while acute triangles have all angles less than 90 degrees, and right triangles have one angle exactly at 90 degrees. This distinction affects their properties significantly; for instance, in an obtuse triangle, the side opposite the obtuse angle is always the longest. Additionally, while an obtuse triangle can be either scalene or isosceles, both acute and right triangles can also fit these classifications but lack the obtuse angle.
Discuss how the Triangle Inequality Theorem applies specifically to obtuse triangles and provide an example.
The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This applies to obtuse triangles just as it does to all triangles. For example, consider an obtuse triangle with sides measuring 3 cm, 4 cm, and 6 cm. Here, if we take any two sides, such as 3 cm and 4 cm, their sum (7 cm) is greater than the third side (6 cm), satisfying the theorem.
Evaluate the implications of having an obtuse angle in a triangle on its geometric properties and possible applications.
Having an obtuse angle in a triangle significantly influences its geometric properties, including side lengths and angle measures. For instance, since one angle is greater than 90 degrees, this necessitates that the remaining two angles are acute, fundamentally altering calculations related to area and perimeter. In practical applications, such as architecture or design, recognizing that certain structures may form obtuse triangles helps in ensuring stability and proper load distribution in triangular frameworks.
Related terms
Acute Triangle: A triangle where all three interior angles are less than 90 degrees.
Right Triangle: A triangle that has one angle equal to 90 degrees.
Triangle Inequality Theorem: A principle stating that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.