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Chapter 3
Polygons
I. Properties of a Polygon
A. A plane figure formed by three or more consecutive segments (sides or laterals)
B. Each side intersects exactly two other sides at its endpoints.
C. These intersections are called vertices.
D. No three consecutive vertices are collinear.
3.1 Basic Polygons
II. Types of Polygons3 sides – triangle 4 sides – quadrilateral5 sides – pentagon6 sides – hexagon 7 sides – heptagon8 sides – octagon9 sides – nonagon10 sides – decagon12 sides – dodecagon20 sides – icosagon n sides – n-gon
III. Convex and Concave Polygons
A. Convex – any two points inside of the polygon can be connected with a segment that is completely inside the polygon.
B. Concave – opposite of convex
IV. Regular Polygons
A. Polygons that are both equilateral and equiangular.
B. Equilateral – all the sides are the same length
C. Equiangular – all the angles are the same measure.
D. Perimeter – the distance around a polygon
Regular Pentagon
Diagonal – a segment connecting two non-consecutive vertices
3.2 Angles of Polygons
I. Polygon Interior Angle Theorem
5 sides -- pentagon
3 triangles x 180
540°
3.2 Angles of Polygons
I. Polygon Interior Angle Theorem
6 sides -- hexagon
4 triangles x 180
720°
Total number of degrees in a polygon = 180(n – 2)
3.2 Angles of Polygons
I. Polygon Interior Angle Theorem
6 sides -- hexagon
4 triangles x 180
720°
110°
122°
95°
103°x
x + 30
II. Regular Polygons
180 (3)
540° 5
108°
Each angle of a regular polygon = 180(n – 2) n
180 (n – 2)
Regular Octagon
180 ( n – 2)
180 ( 6)
1080° 8
135°
III. Polygon Exterior Angle Theorem
108°
108°
108°108°
108°
72°
72°
72°
72°
72°
The sum of the measures of the exterior angles of a polygon is 360°
3.3 Types of Quadrilaterals
I. Parallelogram - quadrilateral with opposite sides that are parallel and congruent
125°
125°55°
55°
Opposite angles are equal in measure
II. Rectangle
A parallelogram with four right angles.
III. Rhombus
Parallelogram with four congruent sides
120°
120°
60°
60°
IV. Square
Parallelogram with four congruent sides and four right angles
A square is a parallelogram, a rectangle, and a rhombus.
V. Trapezoid
A quadrilateral with one pair of parallel sides
3.4 Trapezoids
Parallel sides are called the bases.
Non-parallel sides are called the legs.
I. Special Trapezoids
A. Isosceles Trapezoid
a trapezoid with two congruent sides
70° 70°
110° 110°
Base angles are congruent
I. Special Trapezoids
B. Right Trapezoid
a trapezoid with two right angles
125°
55°
II. Trapezoid Midsegment Theorem
A B
C D
E F
25 cm.
39 cm.
32 cm.
MS = (sum of bases)2
MS = (25 + 39)2
MS = (64)2MS = 32