Triangles

Triangles Theorem:

The measured angles of a triangle sum up to 180°.

Corollary 1

If two angles of a triangle are congruent to two other angles from another triangle then, the third angles are congruent.

Corollary 2

The individual measured angles of an equiangular triangle is 60°.

Corollary 3

In a triangle, at most, there can only be one right angle or obtuse angle.

Corollary 4

In a triangle, the acute angles are complementary.

Theorem:

The measure of any exterior angle in a triangle is equal to the sum of the measures of the two remote interior angles.

Postulate: The SSS Postulate:

If three sides of a triangle is congruent with three sides of another triangle then, the triangles are congruent.

Postulate: The SAS Postulate:

If two sides and the included angle of a triangle are congruent with two sides and the included angle of another triangle then, the triangles are congruent.

Postulate: The ASA Postulate:

If two angles and the included side of a triangle are congruent with two angles and the included side of another triangle then, the triangles are congruent.

Theorem: The Isosceles Triangle Theorem:

If two sides of a triangle are congruent then the angles opposite those sides are congruent.

Corollary 1:

An equilateral triangle is also an equiangular triangle.

Corollary 2:

An equilateral triangle has three 60° angles.

Corollary 3:

The bisector of the vertex angle of an isosceles triangle is perpendicular to the base at its midpoint.

Theorem: The AAS Theorem:

If two angles of a triangle are congruent, then the sides opposite the angles are congruent.

Theorem:

If two angles and the non-included side of a triangle are congruent to two angles and the non-included side of another triangle then , the triangles are congruent.

A median of a triangle is a segment from a vertex to the midpoint of the opposite side of the vertex.

An altitude of a triangle is a segment from a vertex and it is perpendicular to the opposite side of the vertex.

Theorem:

The segment whose endpoints are the midpoints of two sides of a triangle:

a) is parallel to the third side.

b) its length is half the length of the third side.