Notes on Angles of Polygons

Angles of Polygons Overview

• Types of Angles: Interior angles and Exterior angles of polygons.

Theorem 6.1: Polygon Interior Angles Sum

• Formula: The sum of the interior angle measures of an n-sided convex polygon is given by:
  • $S = (n - 2) imes 180$
• Example: For a nonagon (9 sides):
  • Calculation:
    $S = (9 - 2) imes 180 = 7 imes 180 = 1260$

Finding Interior Angle Measures

• Application: To find the measure of each interior angle in a regular polygon:
  • For a polygon with 5 sides (pentagon):
  • Total sum of interior angles:
    $S = (5 - 2) imes 180 = 3 imes 180 = 540$
  • Measure of each angle:
    $ext{Each angle} = \frac{540}{5} = 108$

Theorem 6.2: Polygon Exterior Angles Sum

• Formula: The sum of the exterior angle measures of a convex polygon is always:
  • $S = 360$
• Example: For a polygon with 6 angles
  • Calculation:
    $mZ<em>1 + mZ</em>2 + mZ<em>3 + mZ</em>4 + mZ<em>5 + mZ</em>6 = 360$

Finding Exterior Angle Measures

• Regular Decagon: For a regular decagon (10 sides), each exterior angle can be found as follows:
  • Total sum of exterior angles:
  • Equation:
    $10n = 360$
  • Measure of each angle:
    $n = \frac{360}{10} = 36$

Example Problems

• Find the number of sides using interior angle measures: If an interior angle is $150$:
  • Set the equation using the Interior Angle Sum Theorem:
    $S = 180(n - 2)$
  • Set equal to each angle's measure:
    $150n = 180(n - 2) \ \ 150n = 180n - 360$
  • Rearranging gives:
    $30n = 360 \ \ n = 12$

Vocabulary

• Diagonal: Line segment connecting two non-adjacent vertices in a polygon.
• Convex polygon: A polygon where all interior angles are less than 180 degrees.

Practical Applications

• Used in architecture, design, and various fields needing geometry calculations.
• Understanding angles is crucial for tasks involving construction and spatial understanding.