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Fundamentals of Triangles: Properties, Theorems, and Applications - Prof. Cabug-Us, Study notes of Geometry

A comprehensive overview of the fundamental concepts related to triangles, including their definition, key theorems and properties, and various types of triangles. It covers topics such as the sum of the angles in a triangle, the relationship between side lengths and angles, the properties of similar triangles, and formulas for calculating the area and perimeter of triangles. This information can be valuable for students studying geometry, mathematics, and related fields, as it lays the foundation for understanding more advanced concepts and problem-solving techniques involving triangles. The document could be particularly useful for university-level courses in subjects like euclidean geometry, trigonometry, and discrete mathematics, as well as for high school students preparing for exams or exploring the mathematical properties of triangles.

Typology: Study notes

2022/2023

Available from 08/26/2024

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TRIANGLE
1.1. WHAT IS A TRIANGLE?
A triangle is a polygon with three sides having
three vertices. The angle formed inside the
triangle is equal to 180 degrees.
1.2. THEOREMS AND PROPERTIES
1. The sum of the three angles of a triangle is
equal to two right angles or 180.
2. The sum of two sides of a triangle is greater
than the third side, and their difference is
less than the third side.
3. Two triangles are said to be similar if the
corresponding angles are congruent and the
corresponding side lengths are proportional.
4. If two sides of a triangle are equal (an isosceles triangle),
the angles opposite these sides are equal.
5. If two sides of a triangle are unequal, the angles opposite are
unequal, and the greater angle is opposite the greater side
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TRIANGLE

1.1. WHAT IS A TRIANGLE?

A triangle is a polygon with three sides having three vertices. The angle formed inside the triangle is equal to 180 degrees. 1.2. THEOREMS AND PROPERTIES

  1. The sum of the three angles of a triangle is equal to two right angles or 180.
  2. The sum of two sides of a triangle is greater than the third side, and their difference is less than the third side.
  3. Two triangles are said to be similar if the corresponding angles are congruent and the corresponding side lengths are proportional.
  4. If two sides of a triangle are equal (an isosceles triangle), the angles opposite these sides are equal.
  5. If two sides of a triangle are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side
  1. The perpendicular bisector of the sides, and the bisectors of the angles of a triangle, meet in points which are the center of the circumscribed circle and the inscribed circle, respectively.
  2. The altitudes of a triangle meet in a point (called orthocenter).
  3. The medians of a triangle are concurrent at a point which is two-thirds of the distance from the vertex to the midpoint of the opposite side. The point of concurrency is the centroid of the triangle.

1.3. TYPES OF TRIANGLES

  • Type of triangle according to its angle.
  • Type of triangle according to length of its side.
  • Type of triangle according to both angle and sides. 1.4. FORMULAS REALTING TO TRIANGLE