1. Polygons

2. 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

3. 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.

4. 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.

5. 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

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

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

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

9. Triangles and Quadrilaterals

10. 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

11. 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˚

12. 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.

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

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

15. 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?

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

20. The two theorems below can also be used to show that a given quadrilateral is a parallelogram.

21. 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).

22. Since a rectangle is a parallelogram by Theorem 6-4-1, a rectangle “inherits” all the properties of parallelograms that you learned in Lesson 6-2.

23. Example 1: Craft Application
A woodworker constructs a rectangular picture frame so that JK = 50 cm and JL = 86 cm. Find HM.
Rect.  diags. 
KM = JL = 86
Def. of  segs.
 diags. bisect each other
Substitute and simplify.

24. 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).

26. Like a rectangle, a rhombus is a parallelogram
Like a rectangle, a rhombus is a parallelogram. So you can apply the properties of parallelograms to rhombuses.

27. Example 2A: Using Properties of Rhombuses to Find Measures
TVWX is a rhombus. Find TV.
WV = XT
Def. of rhombus
13b – 9 = 3b + 4
Substitute given values.
10b = 13
Subtract 3b from both sides and add 9 to both sides.
b = 1.3
Divide both sides by 10.

28. Example 2A Continued
TV = XT
Def. of rhombus
Substitute 3b + 4 for XT.
TV = 3b + 4
TV = 3(1.3) + 4 = 7.9
Substitute 1.3 for b and simplify.

29. 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.

30. 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.

31. 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.

32. 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!

33. 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

34. Class Quiz
A walkway 3.0 m wide is constructed along the outside edge of a square courtyard. If the perimeter of the courtyard is 320 m, what is the perimeter of the square formed by the outer edge of the walkway?