Plane Geometry - Plane Figures (Geometric Figures) - Triangles
     Triangles
      Types of triangles
      Main properties of triangles
      Congruence of triangles
         Theorems about congruence
      Similarity of triangles, division of a line segment in a given ratio
         Similarity criteria of triangles
      Area of a triangle
Types of triangles
Types of triangles categorized by their sides are; a scalene triangle, isosceles triangle and equilateral triangle.
The types of triangles categorized by their angles are; an acute triangle, obtuse triangle and right triangle.
Main properties of triangles
1) A sum of triangle angles a + b + g = 180.
2) Angles lying opposite the equal sides are also equal, and inversely.
All angles in an equilateral triangle are also equal. It follows, that each angle in an equilateral triangle is equal to 60 degrees.
3) In any triangle, if one side is extended, the exterior angle is equal to a sum of interior angles, not supplementary.
The sum of exterior angles is 360.
4) Any side of a triangle is less than a sum of two other sides and greater than their difference. 
5) An angle lying opposite the greatest side, is also the greatest angle, and inversely.
Proof: By turning the side BC, of the scalene triangle ABC below, around the vertex C by the angle g into direction of the side AC, obtained is the isosceles triangle BCD with equal angles on the base BD.
Its lateral side CD < AC. Angle a < b', as b' is the exterior angle of the triangle ABD, and b' < b, therefore a < b. Thus, proved is the above statement.
Congruence of triangles
Two figures are called congruent if they have identical size and shape, i.e., if their corresponding angles and sides are equal.
The two congruent figures fit on top of each other exactly. We prove the congruence of two figures by rotation and translation.
Theorems about congruence of triangles are; Two triangles are congruent:
1) If a pair of corresponding sides and the included angle are equal SAS (Side-Angle-Side).
2) If their corresponding sides are equal SSS. 
3) If a pair of corresponding angles and the included side are equal ASA. 
The congruence of two triangles we denote as D ABC @ D A'B'C'.
Similarity of triangles, division of a line segment in a given ratio
Two plane figures are similar if differ in scale not in shape.
Two polygons are similar, if their angles are equal and sides are proportional.
Similarity criteria of triangles are; Two triangles are similar:
1) If all their corresponding angles are equal. 
2) If all their sides are proportional. 
3) If one angle of a triangle is congruent to one angle of another triangle and the sides that include those angles are proportional.
Similarity of two triangles is denoted as D ABC ~ D A'B'C'.
Division of a line segment AB to equal parts: Example:
Division of the line segment AB in a given ratio:  Example:  AC : BC = 1 : 2
Area of a triangle
The area A of any triangle is equal to one-half the product of any base and corresponding height h
A height or altitude of a triangle is a straight line through a vertex and perpendicular to the opposite side.
This opposite side is called the base of the altitude, and the point where the altitude intersects the base (or its extension) is called the foot of the altitude.
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