Properties of Triangles

Triangles are fundamental shapes in geometry, with unique properties that govern their angles and sides. Understanding these properties, like the sum of interior angles and the triangle inequality theorem, is essential for solving various geometric problems.

1. Sum of interior angles is 180 degrees

   • The sum of the angles inside any triangle always equals 180 degrees.
   • This property holds true for all types of triangles: scalene, isosceles, and equilateral.
   • It is fundamental for solving problems involving angle measures in triangles.

2. Triangle inequality theorem

   • The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
   • This theorem ensures that a triangle can be formed with given side lengths.
   • It is crucial for determining the feasibility of constructing a triangle from three given lengths.

3. Exterior angle theorem

   • The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles.
   • This theorem helps in finding unknown angles in a triangle.
   • It reinforces the relationship between interior and exterior angles.

4. Pythagorean theorem

   • In a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides (a² + b² = c²).
   • This theorem is essential for solving problems involving right triangles.
   • It is widely used in various applications, including construction and navigation.

5. Congruence criteria (SSS, SAS, ASA, AAS)

   • SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, the triangles are congruent.
   • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
   • ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) also establish congruence based on angle and side relationships.

6. Similarity criteria (AA, SAS, SSS)

   • AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
   • SAS (Side-Angle-Side): If two sides of one triangle are in proportion to two sides of another triangle and the included angles are equal, the triangles are similar.
   • SSS (Side-Side-Side): If the sides of one triangle are in proportion to the sides of another triangle, the triangles are similar.

7. Median properties

   • A median of a triangle is a line segment from a vertex to the midpoint of the opposite side.
   • Each triangle has three medians, and they intersect at a point called the centroid.
   • The centroid divides each median into a ratio of 2:1, with the longer segment being closer to the vertex.

8. Altitude properties

   • An altitude of a triangle is a perpendicular segment from a vertex to the line containing the opposite side.
   • Each triangle has three altitudes, which may lie inside or outside the triangle depending on the type of triangle.
   • The point where all three altitudes intersect is called the orthocenter.

9. Angle bisector properties

   • An angle bisector is a line segment that divides an angle into two equal angles.
   • The angle bisector of a triangle can be used to find the ratio of the lengths of the two sides adjacent to the angle.
   • The point where the three angle bisectors intersect is called the incenter, which is the center of the triangle's inscribed circle.

10. Centroid, orthocenter, and circumcenter

    • The centroid is the point of intersection of the medians and represents the triangle's center of mass.
    • The orthocenter is the intersection point of the altitudes and can be inside, outside, or on the triangle depending on its type.
    • The circumcenter is the point where the perpendicular bisectors of the sides intersect and is the center of the triangle's circumscribed circle.