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Geometry - Polygons

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    Study Guide Geometry – Polygons 1. Angle Sum of Polygons 1.1 Breaking Polygons into Triangles Figure 1 When you have a polygon with four or more sides , there’s a helpful trick you can use: Draw all possible diagonals from one vertex . When you do this, the polygon is divided into several non - overlapping triangles . This process is called triangulation . Why is this useful? Because we already know something important: The interior angles of a triangle always add up to 180° . So if we know how many triangles are formed, we can find the total interior angle sum of the polygon. 1.2 How Many Triangles Are Formed? If a polygon has n sides , drawing diagonals from one vertex will divide it into: Notice: The number of triangles is always two less than the number of sides .

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    Study Guide Theorem 39: Interior Angle Sum Formula If a convex polygon has n sides , then its interior angle sum is: This formula works for any convex polygon . Example: A 7 - Sided Polygon A 7 - sided polygon is divided into: So the interior angle sum is: 1.3 Exterior Angles of a Polygon Now let’s talk about exterior angles . Figure 2 The (non - straight) exterior angles of a polygon. An exterior angle is formed when you extend one side of a polygon. The angle formed outside the shape (next to the interior angle) is the exterior angle. Each vertex of a polygon has one exterior angle.

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    Study Guide Theorem 40: Exterior Angle Sum If a polygon is convex , then: The sum of the exterior angles, one at each vertex, is always 360° . No matter how many sides the polygon has, the total is: Example 1: Find the Interior Angle Sum of a Decagon A decagon has 10 sides . Use the formula: Example 2: Find the Exterior Angle Sum of a Convex Nonagon A nonagon has 9 sides. But remember: The sum of the exterior angles of any convex polygon is always: So the answer is simply:

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    Study Guide Example 3: Find Each Interior Angle of a Regular Hexagon Figure 3 An interior angle of a regular hexagon. A regular polygon has: • All sides equal • All interior angles equal A hexagon has 6 sides . Method 1: Use the Interior Angle Sum Formula First find the total interior angle sum: Since it’s regular, all 6 angles are equal. Each interior angle measures:

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    Study Guide Method 2: Use Exterior Angles Because the hexagon is regular, all exterior angles are equal. We know the sum of exterior angles is: So each exterior angle is: Since an interior angle and its exterior angle form a straight line: So each interior angle is again: Key Ideas to Remember • A polygon can be divided into ( n - 2 ) triangles. • Interior angle sum formula: S = (n - 2) × 180° • The sum of exterior angles (one per vertex) is always 360° . • In a regular polygon, all angles are equal. • Interior angle + exterior angle = 180°. Summary When finding interior angle sums: 1. Count the number of sides. 2. Use ( (n - 2) × 180°). When finding each interior angle of a regular polygon: 1. Find the total interior sum. 2. Divide by the number of sides. Or: 1. Divide 360° by the number of sides to find each exterior angle. 2. Subtract from 180° to find each interior angle.

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    Study Guide 2. Special Quadrilaterals Not all quadrilaterals are the same. They may look similar, but their properties (special features) are different. Some quadrilaterals: • Have one pair of parallel sides • Have two pairs of parallel sides • Have right angles • Have special relationships between sides and angles 2.1 Trapezoid A trapezoid is a quadrilateral with exactly one pair of opposite sides parallel . Important Vocabulary • The parallel sides are called the bases . • The nonparallel sides are called the legs . • A segment joining the midpoints of the legs is called the median of the trapezoid . • A segment drawn perpendicular to the bases is called the altitude . • The length of the altitude is called the height . Figure 1 A trapezoid with its median and an altitude.

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    Study Guide Understanding the Diagram In the diagram: • AB and CD are the bases (parallel sides). • XY is an altitude (it is perpendicular to the bases). • The length of XY is the height. • MN is the median (it connects the midpoints of the legs). 2.2 Parallelogram A parallelogram is a quadrilateral with both pairs of opposite sides parallel . Each pair of parallel sides is called a pair of bases . A segment drawn perpendicular between a pair of bases is called the altitude of the parallelogram . Because a parallelogram has two pairs of parallel sides , it has two possible heights — one for each pair of bases. The symbol ▱ is often used to represent a parallelogram. Figure 2 A parallelogram with its bases and associated heights. Understanding the Diagram In ▱ ABCD: • XY is an altitude to bases AB and CD . • JK is an altitude to bases AD and BC . • The length of XY is the height when AB and CD are bases. • The length of JK is the height when AD and BC are bases.

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    Study Guide 2.3 Theorems About Parallelograms Parallelograms have many important properties. These are essential for solving geometry problems. Theorem 41 A diagonal of a parallelogram divides it into two congruent triangles . In ▱ ABCD, if diagonal BD is drawn: Figure 3 Two congruent triangles created by a diagonal of a parallelogram. Theorem 42 Opposite sides of a parallelogram are congruent. In ▱ ABCD: Theorem 43 Opposite angles of a parallelogram are congruent.

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    Study Guide Theorem 44 Consecutive (next to each other) angles of a parallelogram are supplementary. That means they add up to: So: • ∠ A and ∠ B are supplementary • ∠ B and ∠ C are supplementary • ∠ C and ∠ D are supplementary • ∠ A and ∠ D are supplementary Figure 4 A parallelogram. Theorem 45 The diagonals of a parallelogram bisect each other . That means they cut each other in half. In ▱ ABCD:

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    Study Guide Figure 5 The diagonals of a parallelogram bisect one another. Summary Trapezoid • One pair of parallel sides • Bases, legs, median, altitude, height Parallelogram • Two pairs of parallel sides • Opposite sides are equal • Opposite angles are equal • Consecutive angles are supplementary • Diagonals bisect each other • A diagonal creates two congruent triangles 3. Proving that Figures Are Parallelograms In geometry, you won’t always be told that a shape is a parallelogram. Often, you will need to prove it using certain properties. Fortunately, there are several theorems that help us test whether a quadrilateral is a parallelogram. Let’s go through them one by one.

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