THE QUADRILATERAL FAMILY

Transcription

1 H UILL FMILY efinitions, Properties hart, Family ree, & Venn iagram Name: ate: Period: Use the textbook (page 306 and chapter 6) to write the definitions. Parallelogram: _ hombus: _ ectangle: _ quare: rapezoid: _ Isosceles rapezoid: Kite: Place a checkmark in the column if the characteristic is always true for each quadrilateral name. nswer the question/directions for the last 3 rows. * Hint: hese columns should have a total of 5 checks 8 checks 8 checks 11 checks 2 checks 5 checks 5 checks Property Parallelogram ectangle hombus quare rapezoid Isos rap Kite Parallel sides oth pairs of opposite sides xactly 1 pair of opposite sides ongruent sides ll sides oth pairs of opposite sides 2 pairs of consecutive sides (but the pairs are not to each other) xactly 1 pair of opposite sides upplementary angles oth pairs of opposite angles are suppl ll pairs of consecutive angles suppl xactly 2 pairs of consecutive angles suppl ongruent angles ll angles or 90 (consecutive sides ) iagonals ymmetry rawing oth pairs of opposite angles xactly 1 pair of opposite angles 2 pairs of consecutive angles (but the pairs are not to each other) iagonals bisect each other iagonals are iagonals are oth diagonals bisect opposite angles xactly 1 diagonal bisects opposite angles xactly 1 diagonal bisects the other diagonal How many lines of symmetry? What degree angle of rotational symmetry? ketch the quadrilateral:

2 he uadrilateral Family ree he uadrilateral Venn iagram Write the names of the quadrilaterals that correspond with sections #1-8. Overlapping circles create sections that have the properties of both circles. lso, a circle that is completely inside a larger circle has all the properties of the larger circle. 1 2 No Parallel ides and 2 Pairs of ongruent ides 3 xactly 1 Pair of Parallel ides 8 Nonparallel sides are congruent 4 2 Pairs of Parallel ides irections: In each of the figures above, write the name of the quadrilateral which corresponds to it. ach of the following should be used exactly once: PLLLOGM, KI, U, UILL, PZOI, NGL, IOL PZOI, and HOMU. xplanation: Following the arrows: he properties of each figure are also properties of the figure which follows it (passing on genes to the children ). eversing the arrows: very figure is also the one which precedes it (shares the last name of the parent ). xtension: Label each figure with markings (congruency marks, parallel, right angles, etc.) that correspond with its definition. 5 4 ight ngles ongruent ides

3 uadrilaterals xactly pair xactly pairs of consecutive sides diagonals xactly pair of opp. angles One diagonal bisects s and the other. sides _ of sides & are suppl, and & are suppl. legs pairs of base angles diagonals he midsegment connects the of the legs, and is to each base. Opp. ides Opp. ides Opp. ngles onsecutive iagonals are are are ngles each are other It has all _ It has right angles It has diagonals properties of a It has all It has all properties properties of a of a It has all properties of a It has four sides It has diagonals iagonals opp.

4 1. reate quare given vertices ( 1, 2) and (5, 4). here are two possible squares. y 3. reate ectangle given vertices (5, 3) and ( 1, 1) and the slope of = 2. y x x 2. reate hombus given vertices (2, 3), ( 2, 4) and y ( 1, 0). x

5 6-3 Notes: Proving that a uadrilateral is a Parallelogram You can show that a quadrilateral is a parallelogram if you can show that one of the following is true. ef: oth pairs opp. ides Parallelogram etermine whether quadrilateral with the given vertices is a parallelogram. xplain. 4. (2,5), (5,9), (6,3), (3,-1) oth pairs of opp. ides Parallelogram iagonals bisect each other Parallelogram oth pairs opp. ngles Parallelogram One pair of opp. ides are both and Parallelogram etermine if each quadrilateral is a parallelogram. Justify your answer (-1,6), (2,-3), (5,0), (2,9) 30 40

6 Geometry Worksheet 6.3 Parallelograms Name ate Period re the following parallelograms? If yes, why? (use one of the five reasons from section 6.3) If no, tell what else would be needed M is the midpoint of and M

7 tate whether the given information is sufficient to support the statement, uadrilateral is a parallelogram. If the information is sufficient, state the reason. 2 1 O and 24. and 17. O = O and O = O 25. and 18. and and 19. and O = O and and and and 29. and 22. and 30. is supplementary to is supplementary to and 2 3

8 Geometry Worksheet ectangles, quares & hombi (6.4) Name ate Period 1. In rectangle, = 2x + 3y, = 5x 2y, = 22, and = 17. Find x and y. In the diagram for problems 2-7, is a rectangle and Z is a parallelogram. 2. If = 2x + 1 and = 3x 1, find x. Z 3. If m = 70, find m Z. Z 4. If m = 35, find m. Z 5. If m = m, find m. 6. If = x 2 and = 4x 6, what is the value Z of x? Z 7. Z = 6x, Z = 3x + 2y, and = 14 x. Find the values of x and y. Is Z a special parallelogram? If so, what kind? Z Use rectangle UV for questions If m 1 = 30, m 2 = 9. If m 6 = 57, m 4 = 10. If m 8 = 133, m 2 = 11. If m 5 = 16, m 3 = 6 V 7 5 K U

9 12. is a rhombus. If the perimeter of = 68 and = 16, find. 13. is a square. If m = x 2 4x, find x. 20. is a square. = 5x + 2y, = 3x y, and = 11. Find x and y. Use rhombus for problems If m F = 28, m =. 27. Given rhombus, = 5x + y 1, = 18, = 8x 2y + 2. Find x and y. 15. If m F = 16x + 6, x =. F 16. If m = 34, m =. 17. If m F = 120 4x, x =. 18. If m = 4x + 6 and m = 12x 18, x =. 19. If m = x 2 6 and m = 5x + 9, x = 22. is a rectangle. Find the length of each diagonal if = 2(x 3) and = x is a rectangle. Find each diagonal if 3c and = 4 c Given square P, = x 2 2x, = 4x 5. Find x,, and. P Given rectangle 24. If X, find m X. X 25. If m = 30 and = 13, find. 26. If m = 45 and = 6.2, find.

10 Geometry W rapezoids and Kites Identify the quadrilateral based on the given information in the diagram or description. Given information includes right angle symbols, congruent segment marks, congruent angle marks, and parallel marks. o NO assume that pictures are drawn to scale. 7. In kite WXYZ, m WXY = 104, and m VYZ = 49. Find each measure. m VZY = m VXW= m XWZ= 8. Find m 1. In kite FGH, m FJ = 25, and m FGJ = 57. Find each measure. m GFJ = 2. Find m Y m JF= m GH= 9. Find K. 10. Find m = 24 and P = 10. Find P. 4. Find. 11. In kite P, m = 12. In kite P, m = 5. In kite P, m = 45, and m = 30. If = 7, find the perimeter of the kite. 6. In kite P, m = 40 and m = 35. If = 7, find the perimeter of the kite. 45, and m = 30. If = 8, find the perimeter of the kite. 40, and m = 35. If = 7, find the perimeter of the kite.

11 etermine whether WXYZ is a parallelogram, a rectangle, a rhombus, or a square for each set of vertices. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. 29. W(5, 6), X(7, 5), Y(9, 9), Z(7, 10) Parallelogram: ectangle: hombus: quare: etermine whether FGH is a parallelogram, a rectangle, a rhombus, or a square for each set of vertices. tate yes or no for each and explain why or why not. how work to support the explanations. For example, if you say the sides are parallel then you need to calculate the slopes. 31. (0, -3), F(-3, 0), G(0, 3), H(3, 0) Parallelogram: ectangle: hombus: quare: 30. W(-3, -3), X(1, -6), Y(5, -3), Z(1, 0) Parallelogram: ectangle: hombus: quare: 32. (2, 1), F(3, 4), G(7, 2), H(6, -1) Parallelogram: ectangle: hombus: quare:

12 uadrilateral Proofs! Name ate_ Pd efinition of : quadrilateral is a parallelogram both pairs of opposite sides are parallel. heorem 6-1: Parallelogram opposite sides are congruent. heorem 6-2: Parallelogram opposite angles are congruent. heorem 6-3: Parallelogram diagonals bisect each other. heorem 6-8: One pair of opposite sides of a quadrilateral is both congruent and parallel parallelogram. 1. Prove that consecutive angles in a are supplementary, without using any of the theorems above. Given: Prove: and are supplementary, and are supplementary, and are supplementary, and and are supplementary. 2. Prove heorem 6-1, without using any of the theorems above. (Hint: draw.) Given: Prove: and. 4. Prove heorem 6-3, without using any of the theorems above except heorem 6-1. Given:, where and intersect at point. Prove: bisects, and bisects Prove heorem 6-8, without using 5 any of the theorems above. Given: and. Prove: is a parallelogram Prove heorem 6-2, without using any of the theorems above. Given: Prove: and

13 efinition of ectangle: It is a rectangle it is a parallelogram with four right angles. efinition of hombus: It is a rhombus it is a parallelogram with four congruent sides. efinition of rapezoid: It is a trapezoid it is a quadrilateral with exactly one pair of parallel sides. ef of Isosc rapezoid: It is an isosceles trapezoid it is a trapezoid whose nonparallel sides are congruent. efinition of Kite: It is a kite it is a quadrilateral with two pairs of adjacent sides congruent and no opposite sides congruent. You may use any theorems (but write them out!) and definitions above to prove the following. You may also use this unnamed shortcut (write it out!): ongruent supplementary angles are right. 6. Prove heorem 6-9 (rhombus each diagonal bisects two angles of the rhombus). Given: is a rhombus. Prove: and Prove heorem 6-16 (isosceles trapezoid diagonals are congruent), using heorem 6-15 (but write it out). Given: is an isosceles trapezoid with. Prove: 7. Prove heorem 6-10 (rhombus diagonals are perpendicular). Given: is a rhombus. Prove: 11. Prove heorem 6-17 (kite diagonals are perpendicular). Given: is a kite with,, and and intersect at point. Prove: Prove hm 6-11 (rectangle diagonals are ). Given: is a rectangle. Prove: 12. Prove this shortcut statement: If angles are congruent and supplementary, then they are right. Given: 1 2, and 1 and 2 are supplementary. Prove: 1 and 2 are right angles.