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.

 

Right Triangles

Theorem:

If one leg and the hypotenuse of a right triangle is congruent with
the hypotenuse and one leg of another right triangle then the right
triangles are congruent.

In right triangles, the side opposite the right angle is called the hypotenuse, and the other sides are called the legs.
 

Theorem:

The midpoint of the hypotenuse of a right triangle is equidistant from
the three vertices.

 

The Pythagorean Theorem

In a right triangle, the square of the hypotenuse is equal to the sum of
the squares of the legs.

c²= a²+ b²

Theorem:

If the square of one side of a triangle is equal to the sum of the
squares of the other sides then the triangle is a right triangle.

Theorem:

If the square of the longest side of a triangle is greater than the sum
of the squares of the other two sides then the triangle is an obtuse
triangle.

Theorem:

If the square of the longest side of a triangle is less than the sum of
the squares of the other two sides then the triangle is an acute
triangle.

 

Special Right Triangles

Isosceles Right Triangle:

The isosceles right triangle is also called the 45-45-90 triangle because of the measures of the angles.

Theorem: The 45-45-90 Triangle:

The hypotenuse of a 45-45-90 triangle is 2^½ times as long as a leg.

Theorem: The 30-60-90 Triangle:

The hypotenuse of a 30-60-90 triangle is twice as long as the short leg and the longer leg is 3^½ times longer than the shorter leg.