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Name _____________________________ Geometry Date _____________ Period _____________ Definitions: Polygons: a ____________________________ straight-sided figure with ____________________________. Quadrilaterals: a polygon with ____________________________ and ____________________________. Note: The sum of _______________ the angles of a quadrilateral is ________________ Ex: What is the missing angle? 70º 70º ? 110º Parallelograms: a quadrilateral where ________________________ are ________________ Properties of a Parallelogram 1) 2) 3) 4) 5) 1 Name _____________________________ Geometry Mark the following on the picture of the parallelogram below: Date _____________ Period _____________ Congruent angles Congruent sides Parallel sides Supplementary angles Diagonals If quadrilateral FRAC is a parallelogram, what congruence statements can you make? R F C A Ex #1) MATH is a parallelogram. Find the values of w, x, y, and z. M wo yo o H 55 A zo T Ex #2) DUCK is a parallelogram. Find the values of w, x, y, and z. D yo 113o o C z U wo K Directions: Determine whether each of the following figures are parallelograms. Example #1) Example #2) 15 F 6 C J R 80° K 7 16 A 100° 100° A P 2 Name _____________________________ Geometry Example #3) Example #4) E 70° 120° 120° M V Date _____________ Period _____________ 105° 102° A N A 78° R K Directions: Find the value of w, x, y, and z assuming each figure is a parallelogram. Example #1) Example #2) y x 105° 31° y° 78° z 44° x° 29 ° Example #3) z° Example #4) 15 z° 73° w° x° y° z° 55° y° x Example #5) Given parallelogram XCVB with <X=10x-9, and <C=5x+9. Find <V and <B. Example #6) In parallelogram ABCD AB = 2x + 5, CD = y + 1, AD = y + 5, and BC = 3x – 4. Find the measures of the sides. 3 Name _____________________________ Geometry Date _____________ Period _____________ Rectangle Properties: 1) 2) 3) 4 Name _____________________________ Date _____________ Geometry Period _____________ Name two properties that hold for all rectangles but not necessarily for all parallelograms. 1) 2) Discussion: Can it be proved that a rectangle is a parallelogram? If so how? Use rectangle ABCD and the given information to solve each problem. #1) BC=12, find AD A #2) DB=18, find DE 1 B 2 3 E 4 D #3) <1=80°, find <4 C Directions: Find the values of x and y assuming RSTV is a rectangle. Example #1) VW=2x+y WS=36 RS=x-y VT=9 Example #2) VR=y TS=x+11 VT=y-3x RS=x+2 S T S T W W V V R R Directions: If ABCD is a rectangle, find the value of x. Example #1) Example #2) AC=x2 DB=6x-8 Angle DAC=4x+8 Angle CAB=5x-8 C D C D E E 5 A B A B Name _____________________________ Geometry Date _____________ Period _____________ Properties of Rhombi 1) 2) 3) Example #1) Use rhombus PQRS and the given information to solve each problem. a. ST = 13, find SQ Q b. m< PRS = 17, find m<QRS P R c. Find m<STR d. If SP = 4x – 3 and PQ = 18 + x, find x S Example #2) Determine whether the following statements are true or false using the diagram of rhombus RSTV with SV=42. S A) RT=42 T P B) PS=21 R V C) RT perpendicular to SV 6 Name _____________________________ Date _____________ Geometry Period _____________ Example #3) Use Rhombus BCDE and the given information to solve each problem a) If m<EBC = 132.6, find m<EBD C b) If m<BDC = 25.9, find m<EDC c) If m< BEC = 2x + 10 and m<CED = 5x – 20, find x B D E 2 d) If m< CBD = 2x + 24, and m< EBD = x , find x. Properties of a Square: 1) 7 Name _____________________________ Geometry Date _____________ Period _____________ Parallelograms Parallelogram Rectangle Rhombus Square opposite sides parallel opposite sides congruent opposite angles congruent consecutive angles supplementary four right angles four congruent sides diagonals bisect each other diagonals congruent diagonals perpendicular diagonals angle bisector Properties of Trapezoids 1) 2) Isosceles Trapezoid: A trapezoid where the __________________________________. Properties of Isosceles Trapezoids 1) 2) Directions: Label the following on a picture of a trapezoid Base Legs Base angles median 8 Name _____________________________ Geometry Directions: Draw a trapezoid with each set of characteristics. A) both bases are shorter than the legs. Date _____________ Period _____________ B) Contains a right angle. C) Has two obtuse angles. Directions: ABCD is an isosceles trapezoid with bases AD and BC. Use the figure and the given information to solve each problem. A) If BA=9, find CD. B B) If AC=4y-5 and BD=2y+3, find AC and BD. A C D C) If Angle BAD=123, find Angle CBA. Example #2) Given trapezoid GHIJ with median KL, find the value of x. G K J 3x-1 10 7x+1 H L I Example #3) Given trapezoid RSTV with median MN, find the value of x. 6x – 3 15 8x + 5 9 Name _____________________________ Date _____________ Geometry Period _____________ Example #3) HJKL is an isosceles trapezoid with bases HJ and LK. If LK = 30 and HJ = 42 find RS. 30 L K R S H J 42 Example #4) HJKL is an isosceles trapezoid with bases HJ and LK. If RS = 17 and HJ = 14 find LK. L K 17 R S H J 14 Example #5) HJKL is an isosceles trapezoid with bases HJ and LK. If RS = x + 5 and HJ + LK = 4x + 6 find RS. L K x+5 R S J H Example #6) HJKL is an isosceles trapezoid with bases HJ and LK. If m<LRS = 66, find m<KSR. L K 66 R S H J 10 Name _____________________________ Geometry Date _____________ Period _____________ Looking at the chart, answer the following questions: a) Are all rectangles quadrilaterals? b) Are all parallelograms rectangles? c) Are all rectangles squares? d) Are all squares rectangles? e) Are all squares parallelograms? Directions: Make a Venn Diagram of all the Quadrilaterals to show how they are related to each other. How do I tell if it’s a Parallelogram? Use the slope formula to find the slope of opposite pairs of sides. If the slope of opposite pairs of sides is the same then it is a parallelogram. How do I tell if it’s a Rectangle? Check for a parallelogram and then follow the steps below. Look at the slope of two consecutive (next to) sides. If the slope is the –reciprocal then it is a rectangle. How do I tell if it’s a Rhombus? Check for a parallelogram and then follow the steps below. 11 Name _____________________________ Date _____________ Geometry Period _____________ Use the distance formula to find the distance of two consecutive sides. If the distance of two consecutive sides is the same then it is a rhombus. How do I tell if it’s a Square? Check for a parallelogram, rhombus and rectangle. Needs to meet all three requirements to be a square. Example #1) A(0,1) B(2,0) C(3,2) D(1,3) 12 Name _____________________________ Geometry Example #2) A(-1,0) B(1,0) C(3,5) D(1,5) Date _____________ Period _____________ Example #3) A(8,11) B(2,3) C(3,7) D(9,15) 13