Lecture Notes: Angle Measures and Polygons

Introduction to Diagonals

• Definition: A diagonal of a polygon is a segment joining two non-consecutive vertices.
• Consecutive vs Non-consecutive: Consecutive vertices are right next to each other; non-consecutive are not.
  • Example: In quadrilaterals and pentagons, diagonals connect non-consecutive vertices.

Theorem 8.1: Polygon Interior Angles

• Interior Angles Sum:
  • Formula: ( (n - 2) \times 180 )
  • Applies only to convex polygons.
  • Example: Hexagon (6 sides)
    • Calculation: ( (6 - 2) \times 180 = 720 ) degrees.

Corollary: Quadrilateral Interior Angles

• Quadrilateral: Always sums to 360 degrees.
  • Example: Hexagon again demonstrates the interior angles formula.

Example Problems

Example 1

• Find the sum of interior angles of a convex polygon with given total sum (e.g., 1260 degrees).
  • Solve for ( n ):
    • Equation: ( (n - 2) \times 180 = 1260 )
    • Solution: ( n = 9 ) (Nonagon).

Example 2

• Find the value of ( x ) in a quadrilateral:
  • Use corollary: All angles sum to 360.
  • Equation: ( x + 71 + 135 + 112 = 360 )
  • Solution: ( x = 42 ) degrees.

Theorem 8.2: Polygon Exterior Angles

• Exterior Angles Sum:
  • Always equals 360 degrees for any convex polygon.
  • Easier to use than the interior angle formula because it's constant.

Example 3

• Find the value of ( x ) in exterior angles:
  • Equation: ( x + 89 + 2x + 85 = 360 )
  • Simplify and solve: ( x = 62 ).

Regular Polygons

• Regular Polygon: Equilateral and equiangular.
  • Example: Regular 15-gon
    • Interior angles: Use ( n - 2 \times 180 ) then divide by 15 for each angle.
    • Exterior angles: Divide 360 by 15.

Calculation

• Interior Angle Calculation:

  • Formula: ( (15 - 2) \times 180 = 2340 )
  • Each angle: ( \frac{2340}{15} = 156 ) degrees.

• Exterior Angle Calculation:

  • Formula: ( \frac{360}{15} = 24 ) degrees.

Conclusion

• Review of theorems and application in problem-solving for convex polygons.
• Importance of understanding regular polygons for calculating individual angles.