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Quadrilaterals Figures with Four Sides All Quadrilaterals 4 sides Parallelograms Two pairs of opposite sides parallel Rhombus Rectangle Four = angles Four = sides Square Four = sides Four = angles Trapezoid A trapezoid is a quadrilateral with exactly one pair of parallel sides. Base angle base Base angle ► leg Base angle leg ► base Base angle All Quadrilaterals 4 sides Parallelograms Trapezoid Two pairs of opposite sides parallel Rhombus Rectangle Four = angles Four = sides Square Four = sides Four = angles Exactly one pair of opposite sides parallel Midsegment of a Trapezoid The midsegment of a trapezoid is a line segment connecting the midpoints of the legs. ► ► ► The midsegment of a trapezoid is parallel to both bases. Its length is the average of the lengths of the bases. Isosceles Trapezoid ► ► If the non-parallel sides are congruent, the trapezoid is an isosceles trapezoid. All Quadrilaterals 4 sides Parallelograms Trapezoid Two pairs of opposite sides parallel Rhombus Rectangle Four = angles Four = sides Square Four = sides Four = angles Exactly one pair of opposite sides parallel Isosceles trapezoid Legs equal Isosceles Trapezoid Theorems • If a trapezoid is isosceles, both pairs of base angles are congruent • If a trapezoid has a pair of congruent base angles, then it is isosceles. Proof: Let ABCD be an isosceles trapezoid with bases AB and CD A B Draw AE parallel to BC By definition, ABCE is a parallelogram because both pairs of opposite sides are parallel D E C AE = BC because opposite sides of a parallelogram are equal. But since ABCD is isosceles, AD = BC. By substitution, AE =AD . This makes ΔADE isosceles. By the BAT, <D = <AED . Since AE and BC are parallel, and <AED and <C are corresponding angles, <AED = <C . By substitution, <D = <C . <A and <D are supplementary, and <B and <C are supplementary because they are two sets of SSI<s between parallel lines. <A = <B because supplements of equal angles are equal. Isosceles trapezoid theorems • A trapezoid is isosceles if and only if its diagonals are congruent. Kite A kite is a quadrilateral with two pairs of consecutive congruent sides. Some people say that the pairs must be distinct. This means the opposite sides aren’t equal. If you allow opposite sides to be equal, then the rhombuses and squares can be part of the kite family. All Quadrilaterals 4 sides Parallelograms Kite Trapezoid Two pairs of opposite sides parallel Two distinct pairs of consecutive sides = Rhombus Rectangle Four = angles Four = sides Square Four = sides Four = angles Exactly one pair of opposite sides parallel Isosceles trapezoid Legs equal Kite theorems • If a quadrilateral is a kite, then its diagonals are perpendicular. • If a quadrilateral is a kite, then at least one pair of opposite angles are congruent. All Quadrilaterals 4 sides Parallelograms Kite Trapezoid Exactly one pair of opposite sides parallel Two pairs of opposite sides parallel Two distinct pairs of consecutive sides = Rhombus Isosceles trapezoid Rectangle Four = angles Four = sides Legs equal Square Four = sides Four = angles Every figure has its own properties PLUS all the properties of the figures above it as you move backwards up the arrows All Quadrilaterals 4 sides Parallelograms Kite Trapezoid Two pairs of opposite sides parallel Two distinct pairs of consecutive sides = Rhombus Rectangle Four = angles Four = sides Square Four = sides Four = angles Extra animated Exactly one pair of opposite sides parallel Isosceles trapezoid Legs equal
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