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Chapter 3: Polygons and Their Relations

The most important theorem of right triangles is Pythagorean Theorem. It states that the sum of the squares of the two legs is equal to the square of the hypotenuse. This theorem is very useful in determine the length of one side of a right triangle once you know the lengths of the other two sides. And this brings us the Converse of the Pythagorean Theorem: If the sum of the squares of the lengths of two sides of a triangle is equal to the square of the length of the third side, then the triangle is a right triangle. We can also modify the Pythagorean Theorem to determine the whether a triangle is acute or obtuse. If the square of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then the triangle is obtuse; If the square of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then the triangle is acute.

There are two special types of right triangles. They are 45-45-90 triangle and 30-60-90 triangle. In a 45-45-90 triangle, the hypotenuse is /2 times as long as a leg. In a 30-60-90 triangle, the hypotenuse is twice as long as the leg opposite the angle of measure 30, and the leg opposite the angle of measure 60 is /3 times as long as the leg opposite the angle of measure 30.

Quadrilaterals are polygons with four sides. The most commonly seen quadrilaterals are rectangles and squares. Both rectangles and squares belong to a special kind of quadrilateral called parallelogram. Parallelogram is a quadrilateral with both pairs of opposite sides’ parallel. The opposite angles of a parallelogram are congruent. Parallelograms also have following properties:

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