Polygons

Polygons Definition: A closed figure formed by line segments so that each segment intersects exactly two others, but only at their endpoints. These figures are not polygons These figures are polygons

Classifications of a Polygon Convex: No line containing a side of the polygon contains a point in its interior Concave: A polygon for which there is a line containing a side of the polygon and a point in the interior of the polygon.

Classifications of a Polygon Regular: A convex polygon in which all interior angles have the same measure and all sides are the same length Irregular: Two sides (or two interior angles) are not congruent.

Polygon Names 3 sides Triangle 4 sides Quadrilateral 5 sides Pentagon Hexagon 7 sides Heptagon 8 sides Octagon 9 sides Nonagon 10 sides Decagon Dodecagon 12 sides n sides n-gon

Regular Polygons Regular polygons have: All side lengths congruent All angles congruent

Area of Regular Polygon Apothem of a polygon: the distance from the center to any side of the polygon.

Area of Regular Polygon We can now subdivide the polygon into triangles.

Triangles and Quadrilaterals

Classifying Triangles by Sides Scalene: A triangle in which all 3 sides are different lengths. BC = 5.16 cm B C A BC = 3.55 cm A B C AB = 3.47 cm AC = 3.47 cm AB = 3.02 cm AC = 3.15 cm Isosceles: A triangle in which at least 2 sides are equal. HI = 3.70 cm G H I Equilateral: A triangle in which all 3 sides are equal. GI = 3.70 cm GH = 3.70 cm

Classifying Triangles by Angles Acute: A triangle in which all 3 angles are less than 90˚. 57 ° 47 76 G H I Obtuse: 108 ° 44 28 B C A A triangle in which one and only one angle is greater than 90˚& less than 180˚

Classifying Triangles by Angles Right: A triangle in which one and only one angle is 90˚ Equiangular: A triangle in which all 3 angles are the same measure.

Classification by Sides with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle scalene isosceles Scalene Isosceles equilateral Equilateral

Classification by Angles with Flow Charts & Venn Diagrams polygons Polygon triangles Triangle right acute equiangular Right Obtuse Acute obtuse Equiangular

What is a Quadrilateral? All quadrilaterals have four sides. They also have four angles. The sum of the four angles totals 360° These properties are what make quadrilaterals alike, but what makes them different?

Parallelogram Two sets of parallel sides Two sets of congruent sides. The angles that are opposite each other are congruent (equal measure).

Rectangle Has all properties of quadrilateral and parallelogram A rectangle also has four right angles. A rectangle can be referred to as an equiangular parallelogram because all four of it’s angle are right, meaning they are all 90° (four equal angles).

Rhombus A rhombus is sometimes referred to as a “slanted square”. A rhombus has all the properties of a quadrilateral and all the properties of a parallelogram, in addition to other properties. A rhombus is often referred to as a equilateral parallelogram, because it has four sides that are congruent (each side length has equal measure).

Square The square is the most specific member of the family of quadrilaterals. The square has the largest number of properties. Squares have all the properties of a quadrilateral, all the properties of a parallelogram, all the properties of a rectangle, and all the properties of a rhombus. A square can be called a rectangle, rhombus, or a parallelogram because it has all of the properties specific to those figures.

Trapezoid Unlike a parallelogram, rectangle, rhombus, and square who all have two sets of parallel sides, a trapezoid only has one set of parallel sides. These parallel sides are opposite one another. The other set of sides are non parallel.

Isosceles Trapezoid One can never assume a trapezoid is isosceles unless they are given that the trapezoid has specific properties of an isosceles trapezoid. Isosceles is defined as having two equal sides. Therefore, an isosceles trapezoid has two equal sides. These equal sides are called the legs of the trapezoid, which are the non-parallel sides of the trapezoid. Both pair of base angles in an isosceles trapezoid are also congruent.

Right Trapezoid A right trapezoid also has one set of parallel sides, and one set of non-parallel sides. A right trapezoid has exactly two right angles. This means that two angles measure 90°. There should be no problem identifying this quadrilateral correctly, because it’s just like it’s name. When you think of right trapezoid, think of right angles!

Quadrilateral Family Tree It’s important to have a good understanding of how each of the quadrilaterals relate to one another. Any quadrilateral that has two sets of parallel sides can be considered a parallelogram. A rectangle and rhombus are both types of parallelograms, and a square can be considered a rectangle, rhombus, and a parallelogram. Any quadrilateral that has one set of parallel sides is a trapezoid. Isosceles and Right are two types of trapezoids. Parallelogram Trapezoid Rectangle Rhombus Isosceles Trapezoid Right Trapezoid Square