Basic Triangle Proofs
Basic triangle proofs are essential mathematical arguments that explore the properties of triangles, often presented in the structured manner of Euclid's proofs, foundational to high school geometry. These proofs typically include a diagram that labels key features, such as points and angles, alongside a statement of the theorem being demonstrated. Each step in the proof is meticulously justified by referencing definitions, axioms, or previously established theorems. Triangles, characterized as three-sided polygons, can be classified based on side lengths—equilateral, isosceles, and scalene—as well as by their angles, leading to classifications like right, obtuse, and acute triangles.
Triangle proofs commonly address the congruence and similarity of triangles, with similarity defined by equal corresponding angles. Noteworthy examples include the Pythagorean theorem, which establishes a relationship between the sides of right triangles, and has been proven in various ways throughout history. While proofs aim for logical rigor, they can sometimes inadvertently lead to incorrect conclusions if certain assumptions are overlooked. A famous example of this is the fallacy related to isosceles triangles, illustrating the complexities involved in geometric reasoning. Overall, basic triangle proofs serve as a critical tool for understanding fundamental geometric principles.
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Basic Triangle Proofs
Basic triangle proofs are mathematical arguments or statements involving the properties of triangles, and taking the form of Euclid's proofs, which are the basis for high school geometry. Such formal proofs usually, for ease of comprehension and organization, include a diagram in which all features referred to in the proof are named (such as points and angles). This is followed by a clear statement of the theorem the proof intends to demonstrate and an ordered list of statements that formalize the reasoning process proving that theorem. Each such statement is backed by a specific reason: either that it is a "given," or that it is a definition, axiom or postulate, or previously proven theorem or formula.
Triangles are three-sided, three-verticed polygons, denoted according to its vertices, such as ΔABC. Any three points in Euclidean geometry determine both a unique triangle and a unique plane, and triangles may be classified according to the relationship of their sides' lengths. Equilateral triangles have three equal sides; isoceles, two equal sides; and scalene, no equal sides. Triangles are also referred to according to their angles. A triangle with an interior angle of 90 degrees is a right triangle, while an obtuse triangle has an interior angle of more than 90 degrees, and an acute triangle has no interior angle of 90 degrees or more.
Overview
Because triangles are so fundamental to geometry, proofs concerning them are common and prominent. Basic triangle theorems include proofs to show the congruence (exact size and shape) of two or more triangles and proofs showing the similarity of triangles. By definition, triangles are similar if each angle of one triangle has the same measure as its corresponding angle on the other triangle. Proofs may also concern finding the center of a given triangle or type of triangle, or demonstrating specific inequalities of triangles, such as the necessity of two sides being unequal when the pair of angles opposite them is unequal.
The most famous triangle proof is that of the Pythagorean theorem, which states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides. The theorem had been known before the sixth century BCE Greek mathematician Pythagoras, but he was the first to record its proof. It has since been proven, both geometrically and algebraically, through dozens of different proofs, as well as being generalized to non-Euclidean spaces and higher-dimensional objects.
Because they take a form in which every step of reasoning is justified, proofs can, ironically, accidentally lead to erroneous conclusions, or be deliberately constructed to prove falsehoods, by skipping over certain assumptions. One of the most famous examples is the fallacy of the isoceles triangle, which "proves" that all triangles are isoceles triangles. Even students who know this to be false often have difficulty finding the error in the proof, which is often attributed to Lewis Carroll.
Bibliography
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Millman, Richard, Peter Shiue, and Eric Brendan Kahn. Problems and Proofs in Numbers and Algebra. New York: Springer, 2015.
Perrin, Daniel. Algebraic Geometry: An Introduction. New York: Springer, 2008.