The Area Postulate
Every polygonal region has a positive number called its area such that 1) congruent triangles have equal areas and 2) the area of a polygonal region is equal to the sum of the areas of its nonoverlapping parts.
The area of a rectangle is the product of its base and altitude
Corollary to Postulate 9
The area of a square is the square of its side.
The area of a right triangle is half the product of its legs.
The area of a triangle is half the product of any base and corresponding altitude.
Corollary to Theorem 39
Triangles with equal bases and equal altitudes have equal areas
The area of a parallelogram is the product of any base and corresponding altitude
The area of a trapezoid is half the product of its altitude and the sum of its bases.
The Pythagorean Theorem
The square of the hypotenuse of a right triangle is equal to the sum of the squares of its legs.
If the Square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is a right triangle.