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

University of Southern Philippines Foundation (USPF)Geometry
Prof. Helbert Cabug-Us

A comprehensive overview of the fundamental concepts and properties of triangles. It covers the definition of a triangle, the key theorems and properties related to the angles, sides, and relationships between triangles. The document delves into the important concepts of similar triangles, including the relationship between their corresponding sides, angles, and areas. Additionally, it explores the special points and lines within a triangle, such as the orthocenter, centroid, and the circumscribed and inscribed circles. This information is crucial for understanding the underlying principles of geometry and can be applied in various fields, including mathematics, engineering, and architecture. The content presented in this document can serve as valuable study material for students at the high school or university level, as well as for lifelong learners interested in exploring the fascinating world of geometric shapes and their properties.

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.