Geometry - Polygons - Quick Guides

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Study GuideGeometry–Polygons1. Angle Sum of Polygons1.1Breaking Polygons into TrianglesFigure 1When you have a polygon withfour or more sides, there’s a helpful trick you can use:Draw all possible diagonals fromone vertex.When you do this, the polygon is divided into severalnon-overlapping triangles. This process iscalledtriangulation.Why is this useful?Because we already know something important:The interior angles of a triangle always add up to180°.So if we know how many triangles are formed, we can find the total interior angle sum of the polygon.1.2How Many Triangles Are Formed?If a polygon hasn sides, drawing diagonals from one vertex will divide it into:Notice:The number of triangles is alwaystwo less than the number of sides.

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Study GuideTheorem 39: Interior Angle Sum FormulaIf a convex polygon hasn sides, then its interior angle sum is:This formula works forany convex polygon.Example: A 7-Sided PolygonA 7-sided polygon is divided into:So the interior angle sum is:1.3Exterior Angles of a PolygonNow let’s talk aboutexterior angles.Figure 2The (non-straight) exterior angles of a polygon.An exterior angle is formed when you extend one side of a polygon. The angle formed outside theshape (next to the interior angle) is the exterior angle.Each vertex of a polygon has one exterior angle.

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Study GuideTheorem 40: Exterior Angle SumIf a polygon isconvex, then:The sum of the exterior angles, one at each vertex, is always360°.No matter how many sides the polygon has, the total is:Example 1:Find the Interior Angle Sum of a DecagonA decagon has10 sides.Use the formula:Example 2:Find the Exterior Angle Sum of a Convex NonagonA nonagon has 9 sides.But remember:The sum of the exterior angles ofany convex polygonis always:So the answer is simply:

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Study GuideExample 3:Find Each Interior Angle of a Regular HexagonFigure 3 An interior angle of a regular hexagon.Aregular polygonhas:•All sides equal•All interior angles equalA hexagon has6 sides.Method 1: Use the Interior Angle Sum FormulaFirst find the total interior angle sum:Since it’s regular, all 6 angles are equal.Each interior angle measures:

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Study GuideMethod 2: Use Exterior AnglesBecause 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 always360°.•In a regular polygon, all angles are equal.•Interior angle + exterior angle = 180°.SummaryWhen 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 Guide2. Special QuadrilateralsNot all quadrilaterals are the same.They may look similar, but theirproperties(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 angles2.1TrapezoidAtrapezoidis a quadrilateral withexactly one pair of opposite sides parallel.Important Vocabulary•The parallel sides are called thebases.•The nonparallel sides are called thelegs.•A segment joining the midpoints of the legs is called themedian of the trapezoid.•A segment drawn perpendicular to the bases is called thealtitude.•The length of the altitude is called theheight.Figure 1 A trapezoid with its median and an altitude.

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Study GuideUnderstanding the DiagramIn the diagram:•AB and CDare the bases (parallel sides).•XYis an altitude (it is perpendicular to the bases).•The length ofXYis the height.•MNis the median (it connects the midpoints of the legs).2.2ParallelogramAparallelogramis a quadrilateral withboth pairs of opposite sides parallel.Each pair of parallel sides is called a pair ofbases.A segment drawn perpendicular between a pair of bases is called thealtitude of the parallelogram.Because a parallelogram hastwo pairs of parallel sides, it hastwo possible heights—one foreach pair of bases.The symbol▱is often used to represent a parallelogram.Figure 2A parallelogram with its bases and associated heights.Understanding the DiagramIn▱ABCD:•XYis an altitude to basesAB and CD.•JKis an altitude to basesAD and BC.•The length ofXYis the height when AB and CD are bases.•The length ofJKis the height when AD and BC are bases.

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Study Guide2.3Theorems About ParallelogramsParallelograms have many important properties. These are essential for solving geometry problems.Theorem 41A diagonal of a parallelogram divides it intotwo congruent triangles.In▱ABCD, if diagonalBDis drawn:Figure 3Two congruent triangles created by a diagonal of a parallelogram.Theorem 42Opposite sides of a parallelogram are congruent.In▱ABCD:Theorem 43Opposite angles of a parallelogram are congruent.

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Study GuideTheorem 44Consecutive (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 supplementaryFigure 4A parallelogram.Theorem 45The diagonals of a parallelogrambisect each other.That means they cut each other in half.In▱ABCD:

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Study GuideFigure 5The diagonals of a parallelogram bisect one another.SummaryTrapezoid• One pair of parallel sides• Bases, legs, median, altitude, heightParallelogram• 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 triangles3. Proving that Figures Are ParallelogramsIn geometry, you won’t always be told that a shape is a parallelogram.Often, you will need toprove itusing 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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