"A triangle is a polygon with three edges and three vertices."
Triangles are one of the most important shapes in geometry, and this topic covers how to classify triangles by their sides and angles, how to find missing angles and side lengths, and the various theorems that apply to triangles.
Angle measurement in degrees: Understanding how to measure angles in degrees is essential when working with triangles.
Triangle types: There are different types of triangles, including acute, obtuse, and right triangles. Understanding the properties of each type is crucial.
Triangle sides: Triangles have three sides, and understanding their lengths, relationships with each other, and angles is important when working with triangles.
Congruent triangles: Congruent triangles are triangles that are the same in shape and size. Understanding how to identify and work with congruent triangles is essential.
Similar triangles: Similar triangles have the same shape but different sizes. Understanding how to identify and work with similar triangles is important when working with trigonometry.
Pythagorean theorem: The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides. Understanding and applying the Pythagorean theorem is essential when working with triangles.
Special triangles: A special triangle is one with specific properties, like the 30-60-90 or 45-45-90 triangle. Learning how to identify and work with special triangles is useful when solving problems in geometry.
Trigonometric functions: Trigonometric functions such as sine, cosine, and tangent are used to compute angles and distances in triangles. Understanding these functions is essential when working with triangles.
Area and perimeter of triangles: Understanding how to calculate the area and perimeter of triangles is essential in geometry.
Law of Sines and Law of Cosines: The Law of Sines and Law of Cosines are formulas used to find the lengths of the sides and the angles of non-right triangles. Understanding how to apply these formulas is crucial when working with triangles.
Equilateral triangle: A triangle with all sides and angles equal.
Isosceles triangle: A triangle with two sides and two angles equal.
Scalene triangle: A triangle with no equal sides or angles.
Acute triangle: A triangle with all angles less than 90 degrees.
Right triangle: A triangle with one angle equal to 90 degrees.
Obtuse triangle: A triangle with one angle greater than 90 degrees.
Isosceles right triangle: A right triangle with two sides equal.
Isosceles obtuse triangle: An obtuse triangle with two sides equal.
Isosceles acute triangle: An acute triangle with two sides equal.
Pythagorean triangle: A right triangle where the lengths of the sides are integers and follow the Pythagorean theorem.
30-60-90 triangle: A triangle with angles measuring 30, 60, and 90 degrees and sides in a specific ratio.
45-45-90 triangle: A triangle with two equal angles measuring 45 degrees and sides in a specific ratio.
Equiangular triangle: A triangle with all angles equal.
Right isosceles triangle: A triangle with both a right angle and two equal sides.
"A triangle with vertices A, B, and C is denoted ∆ABC."
"Any three points, when non-collinear, determine a unique triangle."
"A unique plane (i.e. a two-dimensional Euclidean space) is simultaneously determined."
"In other words, there is only one plane that contains that triangle."
"No, in higher-dimensional Euclidean spaces, this is no longer true."
"This article is about triangles in Euclidean geometry."
"except where otherwise noted."
"A triangle is a polygon with three edges and three vertices."
"A triangle is a polygon with three edges and three vertices."
"the Euclidean plane, except where otherwise noted."
"A triangle with three edges and three vertices cannot exist in a one-dimensional Euclidean space."
"however, in higher-dimensional Euclidean spaces, this is no longer true."
"If the entire geometry is only the Euclidean plane, there is only one plane and all triangles are contained in it."
"Any three points, when non-collinear, determine a unique triangle and simultaneously, a unique plane."
"It is one of the basic shapes in geometry."
"Any three points, when non-collinear, determine a unique triangle."
"Any three points, when non-collinear, determine a unique triangle."
"Two points cannot form a triangle since three vertices are required."
"If the entire geometry is only the Euclidean plane, there is only one plane."