Triangles are polygons with three sides and three angles. They are classified by their angles as well as by their sides. Like other polygons, triangles have two sets of angles: interior angles and exterior angles. Specifically, a triangle’s interior angles always add up to 180°, and the exterior angles add up to 360°.
Triangle: A closed figure made by three line segments intersecting at their endpoints.
Right Triangle: A triangle with a right angle.
Obtuse Triangle: A triangle with an obtuse angle.
Acute Triangle: A triangle with 3 acute angles.
Equiangular Triangle: A triangle whose angles all have the same measure (60°).
Scalene Triangle: A triangle whose sides all have different lengths.
Isosceles Triangle: A triangle with at least two sides of equal length.
Equilateral Triangle: A triangle with three sides of equal length.
Interior Angle: The angle inside of a closed figure with straight sides.
Exterior Angle: The angle formed by one side of a polygon and the extension of the adjacent side.
Corollary: A theorem that follows quickly, easily, and directly from another theorem.
A triangle is any closed figure formed by three line segments intersecting at their endpoints only.
4. All polygons have two sets of exterior angles - one goes around the triangle clockwise, the other goes around counterclockwise.
5. The interior angle and its adjacent exterior angle form a linear pair by definition.
6. The exterior angles are vertical angles and are therefore congruent.
7. If an exterior angle is indicated, the two angles in a triangle that are not adjacent to the exterior angle are called the remote interior angles.
All THREE angles must be acute in an acute triangle.
Equilateral Triangles Theorem: All equilateral triangles are also equiangular. All equiangular triangles are also equi-lateral.
You can classify a triangle by its angles and side lengths (e.g. an obtuse scalene triangle, an acute isosceles triangle).
A triangle is labeled with a \(\triangle\) and its vertices. The order the vertices are listed in does not matter.
There are some terminology associated with right triangles. The sides adjacent to the right angle are called the legs, while the side opposite to the right angle is called the hypotenuse.
Isosceles triangles also have special terminology. The congruent sides of the isosceles triangle are called legs, while the other side is called the base. The angles between the base and the legs are called the base angles. The angle made by the two legs is called the vertex angle.
Triangle Sum Theorem: Sum of interior angles in a triangle is always 180°.
Corollary to the Triangle Sum Theorem: The acute angles of a right triangle are complementary.
Exterior Angle Sum Theorem: Each set of exterior angles of a polygon add up to 360°.
Exterior Angle Theorem: Sum of the remote interior angles is equal to the non-adjacent exterior angle.