Polygons and Quadrilaterals By: Ou Suk Kwon

Polygons A Polygon is any figure that has more than 3 sides closed, and they are not curved. The exterior angles of every polygon always add up to 360 degrees. To get the interior angles of a regular polygon, equiangular and equilateral, is (n-2) times 180. measure of each angle and n being the amount of sides that a polygon has. A convex polygon is a polygon with all vertices pointing out and can be regular or irregular. A concave polygon is one that has one or more vertices pointing in so it cannot be a regular polygon. An equiangular polygon has all angles equal but the sides can have different size compared to each other. The equilateral is the opposite. It always has congruent sides but the angles can be anything they want as long as the sides are congruent and the figure is closed.

Examples of polygons

Interior Angle Theorem The interior angle theorem for polygons is that (n-2) times 180 is equal to the total amount of angles in the polygon. N represent the # of the sides that a polygon contains, as the # of sides of the polygons changes, the measurement of the interior angles are different in every different types of polygons. Triangle, quadrilateral, pentagon, hexagon, heptagon, nonagon, decagon, dodecagon and so on.

Interior angles (3-2)180=180 (6-2)180=720 (4-2)180=360

4 theorems of parallelograms Theorem 6-2-1 says If a quadrilateral is a parallelogram then its opposite sides are congruent. Converse: If quadrilateral’s opposite sides are congruent then it is a parallelogram. Theorem 6-2-3 says if a quadrilateral is a parallelogram then its consecutive angles are add up to 180 (supplementary). Converse: If a quadrilateral’s consecutive angle are supplementary then it is a parallelogram. Theorem 6-2-2 says If a quadrilateral is a parallelogram then its opposite angles are congruent. Converse: If a quadrilaterals opposite angles are congruent then it is a parallelogram. Theorem 6-2-4 says If a quadrilateral is a parallelogram then its diagonals bisect each other. Converse: If a quadrilaterals diagonals bisect each other then it is a parallelogram.

examples

how to prove that a quadrilateral is a parallelogram There are 6 ways to prove that a quadrilateral is parallelogram. 1) Both opposite sides are congruent 2) Both opposite angles are congruent 3) Both opposite sides are parallel 4) Consecutive angles are supplementary 5) One pair of congruent and parallel sides 6) When the diagonals bisect

Rhombus, Square and Rectangle A rhombus has its own characteristics, same as rectangle, but a square is both a rhombus and a rectangle because of its unique and special characteristics. A rhombus is a parallelogram where all sides are congruent and the diagonals are perpendicular. A rectangle has all angles that are right angles, and the diagonals are congruent. A square is equiangular and equilateral, the diagonals are congruent and are perpendicular. Square can be parallelogram, rhombus, and rectangle. But by the converse we can’t say.

example

Trapezoids a trapezoid has one pair of parallel sides. They have two bases and two legs but the legs are never parallel to each other. There is a special trapezoids, called isosceles trapezoids, and this is where opposite angles are supplementary, the legs are congruent, the base angles are congruent and the diagonals bisect. Theorem 6-6-3: If a quadrilateral is an isosceles trapezoid, then each pair of base angles are congruent. Theorem 6-6-4: If a trapezoid has one pair of congruent base angles, then the trapezoid is isosceles. To find midsegment you have to use (B1+B2)/2

Examples

Kites! A kite is a quadrilateral that has two pair of congruent adjacent sides, the diagonals are perpendicular, the have on pair of congruent angle which is where the two non congruent sides meet and one diagonal is bisected which is the one going from non congruent to non congruent adjacent sides. The theorems are: 6-6-1 theorem: if a quadrilateral is a kite, then is diagonals are perpendicular. Converse: if diagonals are perpendicular, then the quadrilateral is a kite. 6-6-2 theorem: if a quadrilateral is a kite, then exactly one pair of opposite angles are congruent. Converse: if exactly of pair of opposite angles are congruent, then the quadrilateral is a kite.

Examples