Geometry: Polygons and Quadrilaterals

Interior Angles of Polygons

• Formula for sum of one interior angle: ((n-2)180/n).
• To find the number of sides (n) of a polygon, set the sum of interior angles equal to ((n-2)180).

Parallelogram and its Properties

• A parallelogram is defined as a quadrilateral with both pairs of opposite sides parallel.
• To find the intersection point in a parallelogram, apply the midpoint formula to the coordinates of a diagonal.

Midpoint and Slope Formulas

• Midpoint formula: ((x₁+x₂)/2, (y₁+y₂)/2).
• Slope formula: (m = (y₂-y₁)/(x₂-x₁)).

Finding Points and Sides

• To find coordinates of an unknown point in a figure, calculate the slope between known points and apply it to other coordinates.

Identifying Quadrilaterals

• Use the Distance Formula to prove congruency of sides and the Slope Formula to confirm parallel lines when identifying parallelograms.
• Distance formula: (d = \sqrt{(x₂ - x₁)² + (y₂ - y₁)²}).

Types of Quadrilaterals

• Rhombus: A parallelogram with four congruent sides.
• Rectangle: A quadrilateral with four right angles.
• Square: A parallelogram that combines properties of both rhombus and rectangle.

Characteristics of Special Quadrilaterals

• Rhombus: Diagonals are perpendicular and bisect a pair of opposite angles.
• Rectangle: Diagonals are congruent.
• Square: Diagonals are both perpendicular and congruent.

Trapezoids

• A trapezoid has one pair of parallel sides.
• Base angles: Two consecutive angles sharing a base.
• Legs: Non-parallel sides of a trapezoid.

Identifying Trapezoids

• Check slopes of opposite sides to determine parallelism.
• Prove legs are congruent using the Distance Formula to identify isosceles trapezoids.

Trapezoid Mid Segment

• The midsegment of a trapezoid is calculated as (1/2(b₁ + b₂)).

Kites

• A kite has two pairs of consecutive congruent sides with no pairs of opposite sides being congruent.
• Characteristics include one pair of opposite angles being congruent and perpendicular diagonals.

Finding Rectangles on a Coordinate Plane

• To identify a rectangle, verify the slopes of pairs of opposite sides; they must be negative reciprocals indicating perpendicularity.