Polygon Theorems Study Guide

Definition of Polygons: A polygon is a closed two-dimensional geometric figure formed by a finite number of straight line segments that intersect only at their endpoints.
Polygon Angle Interior Angle Sum Theorem:
- This theorem provides a formula to calculate the total degrees in a polygon based on the number of sides it has.
- Formula: $180(n - 2)$ where:
- $n$ = number of sides of the polygon.
Convex vs Concave Polygons:
- Convex Polygon: All vertices point outward.
- Concave Polygon: At least one vertex points inward, toward the center of the figure.
Examples Using the Formula:
- Triangle:
- Number of sides: 3
- Calculation: $180(3 - 2) = 180$ degrees.
- Rectangle:
- Number of sides: 4
- Calculation: $180(4 - 2) = 360$ degrees.
- Heptagon (7 sides):
- Calculation: $180(7 - 2) = 900$ degrees.
Application Example:
- Building a corner cabinet in a rectangular room requires checking if the angle at the corner is $90$ degrees:
- For a polygon with 5 sides, total interior angles = $180(5 - 2) = 540$ degrees.
- To find the required angle: Subtract known angles from the total to find the unknown angle needed for fitting.
Workbook Exercise:
- Question 2 (p.132): Find the interior angle sum of a 17-sided polygon:
- Calculation: $180(17 - 2) = 2700$ degrees.
- Regular Ingot Problem:
- Given angle: 172.8 degrees; calculation involves total degrees divided by number of sides to find regular polygon's criteria.

Solving the Regular Angle Problem:
- To eliminate fraction in an equation involving angles, multiply by $n$ :
- Example setup: $180n - 360 = 172.8n$ .
Rearranging and Manipulating the Equation:
- Result: Subtract known values accordingly to isolate $n$ .
- Final step yields a polygon with $50$ sides.
Angle Sum Example:
- For a polygon with defined angles, use known angles to calculate unknown ones by finding the total from the polygon and subtracting known values.
- Handling polygons with varying angle measures involves finding remaining angles through division or subtraction based on whether they are known as right angles or congruent angles.

Exterior Angle Theorem:
- Regardless of the polygon type, the sum of all exterior angles equals $360$ degrees.
- Example:
- For a triangle with angles adding to 180 degrees, exterior angles calculated using complementary relationships result in consistent values regardless of the polygon formed.
Practical Application Example:
- Interior angles labeled (1, 2, 3, 4, 5) directly influence exterior angles derived from adjacent calculations.
- Aligning the formulation of complete angular output against known measures confirms theoretical calculations derived from formulas.

Representing Congruent Angles:
- Label angles with variables (e.g., $x$ ) for congruencies and establish relationships for calculations (e.g., $x + 30$ ).
Setting Up Equation:
- Total of labelled angles should match the expected degree total (e.g., $540$ degrees for pentagons).
- Solve the polynomial to isolate $x$ and thus find other angle measures.
Example:
- Calculate angle measures based on labels assigned corresponding to respective properties of the angles to yield conformity with polygon's theoretical output.

Finding Actual Angles from Algebraic Relationships:
- Establish relationships of angles with given algebraic conditions, confirming against total degree expectations from polygon shapes.
Solving Example:
- Measure of a regular 100-gon defined as $3x + 26.4$ leads to solving against derived measures for consistent degrees.

Calculating Interior Angles via Formulas:
- Determine the degree measure of each angle in larger polygons by finding the total first and then balancing differing unknown measures to derive variable clues.

Final Revisions and Confirmations:
- Make sure the total degrees align with theoretical values expected from geometric principles. All interior angle measures must contribute to their overall totals, rechecking arithmetic where necessary.