6.1

Polygon Interior Angle-Sum Theorem

  • Triangle Decomposition

    • Any convex polygon with nn sides can be decomposed into n−2n - 2 non-overlapping triangles by drawing all diagonals from a single vertex.

    • Because the sum of the interior angle measures of each triangle is 180∘180^{\circ}, the sum of the interior angle measures of the polygon is:
          Sum of Interior Angles=180∘(n−2)\text{Sum of Interior Angles} = 180^{\circ}(n - 2)

  • Regular Polygons

    • A regular polygon is both equilateral (all sides congruent) and equiangular (all interior angles congruent).

    • The measure of each interior angle of a regular nn-gon is:
          Measure of each interior angle=180∘(n−2)n\text{Measure of each interior angle} = \frac{180^{\circ}(n - 2)}{n}

Polygon Exterior Angle-Sum Theorem

  • Exterior Angle Definition

    • An exterior angle is formed by extending one side of a polygon past the adjacent vertex.

    • At each vertex, an interior angle and its adjacent exterior angle form a linear pair, which means their measures add up to 180∘180^{\circ}.

  • Sum of Exterior Angles

    • The sum of the measures of the exterior angles of any convex polygon (taking one exterior angle per vertex) is always constant:
          Sum of Exterior Angles=360∘\text{Sum of Exterior Angles} = 360^{\circ}

  • Regular Polygons

    • The measure of each exterior angle of a regular nn-gon is:
          Measure of each exterior angle=360∘n\text{Measure of each exterior angle} = \frac{360^{\circ}}{n}

  • I E Relationships

    • The relationships between the interior angle (I) and the exterior angle (E) is given by I + E = 180degrees.


Key Problem-Solving Strategies

  • Finding the Number of Sides (nn)

    • Given an exterior angle measure (EE) of a regular polygon:
          n=360∘En = \frac{360^{\circ}}{E}

    • Given an interior angle measure (II) of a regular polygon, first determine the exterior angle measure:
          E=180∘−IE = 180^{\circ} - I
          Then solve for nn using the exterior angle formula.

    • Given the sum of interior angles (SS):
          n=S180∘+2n = \frac{S}{180^{\circ}} + 2

  • Solving for Unknown Angles and Variables

    • Interior Angle Problems: Add all given interior angle expressions, set the expression equal to 180∘(n−2)180^{\circ}(n - 2), and solve for the unknown variable.

    • Exterior Angle Problems: Add all given exterior angle expressions, set the sum equal to 360∘360^{\circ}, and solve for the unknown variable.