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Geometry Polygons Lesson 8-4: Parallelograms, Trapezoids and Kites proofs Objectives: Students will be able apply definitions and theorems all parallelograms and kites and trapezoid proofs. Procedure: HW Discussion Quiz: polygon formulas Explore the kite (Math open reference) Note organizer: problems/example proofs Practice proofs Homework: All quad proofs WS 8-4 Mathopenref.com Quadrilaterals Kite Definition: A QUADRILATERAL WITH 2 DISTINCT PAIRS OF CONSECUTIVE CONGRUENT SIDES Draw AND Label diagram of kite: (include diagonals) A kite is a member of the quadrilateral family, and while easy to understand visually, is a little tricky to define in precise mathematical terms. It has two pairs of equal sides. Each pair must be adjacent sides (sharing a common vertex) and each pair must be distinct. That is, the pairs cannot have a side in common. Properties of a kite: 1. Diagonals – are perpendicular -One diagonal bisects the other 2. Angles – angles between the unequal sides are equal 3. Area - d1 d 2 2 where d1 and d2 are the diagonals 4.Perimeter - the sum of the sides Geometry Lesson 8-4: parallelograms, trapezoids, kites Ways to prove each quadrilateral! Rectangle Rhombus Square If a parallelogram contains a right angle, then it is a rectangle. If a quadrilateral is equiangular, then it is a rectangle. If a parallelogram is a rhombus, then it has two consecutive congruent sides. If a quadrilateral is equilateral, then it is a rhombus. If a rectangle is a square, then it has two consecutive congruent sides If a rhombus is a square, then it has a right angle If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. If the diagonals of a parallelogram are perpendicular to each other, then the parallelogram is a rhombus. Area= s2 Area = length x width Area= d1 d 2 2 Trapezoid Kite If a quadrilateral is a trapezoid then it has exactly two parallel sides. If a quadrilateral is a kite then it has two pairs of distinct adjacent congruent sides Properties: b1 b2 h 2 Area= Right trapezoid If a trapezoid is a right trapezoid then it has a right angle. Isosceles trapezoid A trapezoid is isosceles if and only if the diagonals are congruent If a trapezoid is an isosceles trapezoid then the nonparallel sides are congruent. If a trapezoid is an isosceles trapezoid then its opposite angles which are supplementary. A trapezoid is isosceles if and only if the base angles are congruent. Diagonals are perpendicular angles between the unequal sides are equal d1 d 2 Area: 2 Median of a trapezoid The median (also called the mid-segment) of a trapezoid is a segment that connects the midpoint of one leg to the midpoint of the other leg. ; Theorem: The median (or mid-segment) of a trapezoid is parallel to each base and its length is one half the sum of the lengths of the bases. (True for ALL trapezoids.) Examples: 1) EF is the median (mid-segment) of trapezoid ABCD. EF = 25 and AD = 40. Find BC. 40 x 2 50 40 x 25 x 10 BC 10 2) 2 x 3 2 x 11 2 28 4 x 8 4 x 20 x5 PQ 21 14 Example proofs: Statements Reasons 1. 1. Given 2. AM BM 2. Def. of midpoint 3. A B 3. Rectangles are equiangular. 4. DA CB 5. DAM CBM 4. Rectangles have opposite congruent sides. 5. SAS SAS 6. DM CM 6.CPCTC Statements Reasons 1. 1. Given 2. EB CB 2. In a triangle, if 2 angles are equal, their opposites sides are equal 3. AB CB 4. ABCD is a rhombus 3. Substitution 4. A parallelogram with two consecutive congruent sides is a rhombus. Given: LORI is a rectangle BIRD is a trapezoid with bases BD and IR LD OB Prove: BIRD is an isosceles trapezoid Statements Reasons 1. LORI is a rectangle LD OB ; BIRD is a trapezoid 1. Given 2. L O 2. A rectangle is equiangular. 3. LI OR 3. Opposite sides of a rectangle are congruent. 4. LB + BD = LD; BD + DO = BO 4. Partition 5. LB + BD = BD + DO 5. Substitution 6. BD = BD 6. Reflexive 7. LB = OD 7. Subtraction 8. BLI DOR 9. BI DR 10. BIRD is an isosceles trapezoid 8. SAS SAS 9.CPCTC 10. A trapezoid with nonparallel congruent sides is isosceles. Given: PQRS is a kite. Prove: QTR STR Statements Reasons 1. PQRS is a kite. 1. Given 2. PR QS 2. Kites have perpendicular diagonals 3. QTR STR 3. Right angles are congruent 4. RT RT 4. Reflexive 5. ST QT 5. The diagonal that joins the non-congruent sides of a kite is bisected by the other diagonal 6. SAS SAS 6. QTR STR