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Unit 6
Special Quadrilaterals
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Polygon Angle Sum
• The sum of the measures of the interior angles of an n-gon is (n-2)180, where n represents the number of sides
• Example – – What is the sum of the interior angles of a 13
sided figure• (13-2) 180• (11)(180)• 1980°
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You Try
• Sum of the interior angles of a Heptagon– 900°
• Sum of the interior angles of a 17-gon?– 2700°
• Sum of the interior angles of a Quadrilateral? – 360°
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Types of Polygons
• Equilateral Polygon – All sides are congruent
• Equiangular Polygon – All angles are congruent
• Regular polygon – All angles and all sides are congruent
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Finding One Interior Angle of a Regular Polygon
• The measure of each interior angle of a regular n-gon is [(n-2)180] ⁄ n, where n represents the number of sides
• Example:– Find the measure of one interior angle of a regular
hexagon• [(6-2)180]/6• [(4)180]/6• 720/6• 120°
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You Try• Find the measure of one interior angle of a regular 16 –
gon.• 157.5°
• Find the measure of one interior angle of a regular nonagon.• 140°
• Find the measure of one interior angle of a regular 11 – gon.• 147.2727272727
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Polygon Exterior Angle Theorem
• The sum of the measures of the exterior angles of a polygon, with one angle at each vertex, is always 360°.
• Example:– What is one exterior angle of a regular octagon? • 360/8 • 45°
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Parallelograms
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Parallelogram
• A parallelogram is a special quadrilateral with both pair of opposite sides parallel.
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Properties of a parallelogram
• Both pairs of opposite sides are congruent
• Both pairs of opposite angles are congruent
• Consecutive angles are supplementary
• Diagonals Bisect Each other
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How to prove a quadrilateral is a parallelogram
• There are 6 ways to prove that a quadrilateral is a parallelogram
• By the Definition of a parallelogram that
states if both pairs of opposite sides are parallel then a quadrilateral is a parallelogram
• If both pairs of opposite sides are congruent then quadrilateral is a parallelogram
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• If both pairs of opposite angles are congruent then quadrilateral is a parallelogram
• If consecutive angles are supplementary then quadrilateral is a parallelogram
• If diagonals bisect each other then quadrilateral is a parallelogram
• If one pair of opposite sides are congruent and parallel then quadrilateral is a parallelogram
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Rectangle, Rhombus, Square
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Rectangle
• A rectangle is a parallelogram with four right angles.
• If a parallelogram is a rectangle then all parallelogram properties apply.
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Properties of Rectangles
• If a parallelogram is a rectangle then the diagonals are congruent.
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Rhombus
• A Rhombus is a parallelogram with four congruent sides
• If a parallelogram is a rhombus then all parallelogram properties apply.
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Properties of a Rhombus
• If a parallelogram is a rhombus, then its diagonals are perpendicular
• If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles
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Square
• A square is a parallelogram with four congruent sides and four right angles.
• A square is a parallelogram, rectangle, and rhombus so all the properties apply!
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Proving a Quadrilateral is a Rectangle
• To prove a quadrilateral is a rectangle you must first prove it is a parallelogram.
• Then:– If all angles are right angles then parallelogram is
a rectangle.
– If the diagonals of the parallelogram are congruent then it is a rectangle.
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Proving a Quadrilateral is a Rhombus
• To prove a quadrilateral is a rhombus you must first prove it is a parallelogram.
• Then:– If all sides are congruent then parallelogram is a rhombus.
– If the diagonals of the parallelogram are perpendicular then it is a rhombus.
– If the diagonals of a parallelogram bisect opposite angles then it is a rhombus.
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Proving a Quadrilateral is a Square
• To prove a quadrilateral is a square you must first prove it is a parallelogram.
• Then: – Prove that parallelogram is a rhombus.
• Then:– Prove that parallelogram is a rectangle.
Then quadrilateral is a Square!
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Trapezoid / Kite
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Trapezoid
• A trapezoid is a quadrilateral with one pair of parallel sides, know as the bases. The non-parallel sides are know as the legs.
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Isosceles Trapezoid
• A trapezoid with legs that are congruent.
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Properties of Isosceles Trapezoid
• If a quadrilateral is an isosceles trapezoid then each pair of base angles are congruent
• If a quadrilateral is an isosceles trapezoid then diagonals are congruent
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Trapezoid Midsegment Theorem
• The midsegment of a trapezoid is parallel to the bases and is half the sum of the lengths of the bases.
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Kite
• A kite is a quadrilateral with two pairs of consecutive sides congruent and no opposite sides congruent.
• If a quadrilateral is a kite then the diagonals are perpendicular.
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