Squares and Rectangles

Transcription

1 Lesson.1 Skills Practice Name Date Squares and Rectangles Properties of Squares and Rectangles Vocabulary Define the term in your own words. 1. Explain the Perpendicular/Parallel Line Theorem in your own words. Problem Set Use the given statements and the Perpendicular/Parallel Line Theorem to identify the pair of parallel lines in each figure. 1. Given: l m and l r 2. k b and g b l k g h b m p r m r 3. Given: z g and z p 4. Given: c t and t e z g i t c v e b p Chapter Skills Practice 62

2 Lesson.1 Skills Practice page 2 5. Given: n r and r q 6. Given: b x and k b y n q v b x r k e z Complete each statement for square GKJH. G K E H. GK > KJ > JH > HG J _ 8. /KGH > / > / > / > / > / > / > / 9. /GEK, /, /, /, /, /, /, and / are right angles. 10. GK and _ GH 11. GE > > > 12. / KGE > / > / > / > / > / > / > / 628 Chapter Skills Practice

3 Lesson.1 Skills Practice page 3 Name Date Complete each statement for rectangle TMNU. M N L T _ 13. MN _ > TU and MT > NU U 14. /NMT > / > / > / 15. /MTU, /, /, and / are right angles. 16. _ MN 1. _ MU > and MT 18. ML > > > Construct each quadrilateral using the given information. 19. Use AB to construct square ABCD with diagonals AC and BD intersecting at point E. A B D C A E B Chapter Skills Practice 629

4 Lesson.1 Skills Practice page Use _ WX to construct square WXYZ with diagonals _ WY and XZ intersecting at point P. W X 21. Use OP to construct square OPQR with diagonals _ OQ and PR intersecting at point T. O P 630 Chapter Skills Practice

5 Lesson.1 Skills Practice page 5 Name Date 22. Use RE to construct rectangle RECT with diagonals RC and ET intersecting at point A. Do not construct a square. R E 23. Use QR to construct rectangle QRST with diagonals QS and RT intersecting at point P. Do not construct a square. Q R Chapter Skills Practice 631

6 Lesson.1 Skills Practice page Use _ MN to construct rectangle MNOP with diagonals _ MO and NP intersecting at point G. Do not construct a square. M N Create each proof. 25. Write a paragraph proof to prove AC > BD in square ABCD. A B D C You are given square ABCD with diagonals AC and DB. By definition of a square, /ADC and /BCD are right angles. Because all right angles are congruent, you can conclude that /ADC > /BCD. Also by definition of a square, AD > BC. By the Reflexive Property, DC > DC. Therefore, n ADC > n BCD by the SAS Congruence Theorem. So, AC > BD by CPCTC. 632 Chapter Skills Practice

7 Lesson.1 Skills Practice page Name Date 26. Write a paragraph proof to prove GL > IL and HL > JL in square GHIJ. G H L J I 2. Create a two-column proof to prove ML NP in square MNLP. M N Q P L Chapter Skills Practice 633

8 Lesson.1 Skills Practice page Create a two-column proof to prove RT > SU in rectangle RSTU. R S V U T 634 Chapter Skills Practice

9 Lesson.1 Skills Practice page 9 Name Date 29. Create a two-column proof to prove LK > KN and OK > _ KM in rectangle LMNO. L M K O N 30. Write a paragraph proof to prove opposite sides are parallel in rectangle WXYZ. W X A Z Y Chapter Skills Practice 635

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11 Lesson.2 Skills Practice Name Date Parallelograms and Rhombi Properties of Parallelograms and Rhombi Vocabulary 1. Explain how the Parallelogram/Congruent-Parallel Side Theorem can be used to determine if a quadrilateral is a parallelogram. Problem Set Complete each statement for parallelogram MNPL. M N R L P 1. MN _ > LP and ML > NP 2. /NML > / and /MLP > / 3. _ MN 4. _ MR > and ML and LR > Chapter Skills Practice 63

12 Lesson.2 Skills Practice page 2 Complete each statement for rhombus UVWX. U V E X _ W _ 5. UV > VW > WX > XU 6. /UVW > / and /XUV > /. UV 8. UE > and UX and XE > Construct each quadrilateral using the given information. 9. Use RS to construct parallelogram RSTU with diagonals RT and SU intersecting at point G. R S U T G R S 638 Chapter Skills Practice

13 Lesson.2 Skills Practice page 3 Name Date 10. Use AB to construct parallelogram ABCD with diagonals AC and BD intersecting at point F. A B 11. Use _ WX to construct parallelogram WXYZ with diagonals _ WY and XZ intersecting at point H. W X 12. Use AB to construct rhombus ABCD with diagonals AC and BD intersecting at point K. Do not construct a square. A B Chapter Skills Practice 639

14 Lesson.2 Skills Practice page Use _ WX to construct rhombus WXYZ with diagonals _ WY and XZ intersecting at point V. Do not construct a square. w X 14. Use EF to construct rhombus EFGH with diagonals EG and FH intersecting at point T. Do not construct a square. E F Determine the missing statement needed to prove each quadrilateral is a parallelogram by the Parallelogram/Congruent-Parallel Side Theorem. 15. XY _ ZW Z W 16. PS > QR X Y Q _ XY > ZW P S R 640 Chapter Skills Practice

15 Lesson.2 Skills Practice page 5 Name Date 1. _ MN > PQ 18. CF DE M C D P F E N Q 19. ST VU 20. KG > HL S T K G H V U L Create each proof. 21. Write a paragraph proof to prove /BAD > /DCB in parallelogram ABCD. A B D C We are given parallelogram ABCD with diagonals AC and DB. By definition of a parallelogram, AB i CD and AD i BC. So, /ABD > /CDB and /ADB > /DBC by the Alternate Interior Angle Theorem. By the Reflexive Property, DB > DB. Therefore, nabd > ncdb by the ASA Congruence Theorem. So, /BAD > /DCB by CPCTC. Chapter Skills Practice 641

16 Lesson.2 Skills Practice page Create a two-column proof to prove GL > IL and JL > HL in parallelogram GHIJ. G H L J I 23. Write a paragraph proof to prove opposite sides are congruent in parallelogram HIJK. H I K J 642 Chapter Skills Practice

17 Lesson.2 Skills Practice page Name Date 24. Create a two-column proof to prove RT SU in rhombus RSTU. R S M U T Chapter Skills Practice 643

18 Lesson.2 Skills Practice page Create a two-column proof to prove /QPR > /SPR, /QRP > /SRP, /PSQ > /RSQ, and /PQS > /RQS in rhombus PQRS. P Q A S R 644 Chapter Skills Practice

19 Lesson.2 Skills Practice page 9 Name Date 26. Create a two-column proof to prove rhombus RSTU is a parallelogram. R S U T Use the given information to answer each question. 2. Tommy drew a quadrilateral. He used a protractor to measure all four angles of the quadrilateral. How many pairs of angles must be congruent for the quadrilateral to be a parallelogram? Explain. Opposite angles are congruent in a parallelogram, so both pairs of opposite angles must be congruent. 28. Khyree cut a quadrilateral out of a piece of cardstock, but is not sure if the figure is a parallelogram or a rhombus. He measures the lengths of the opposite sides and determines them to be congruent. He measures the opposite angles of the quadrilateral and determines them also to be congruent. He measures one angle and is able to determine that the quadrilateral is a rhombus. What angle did he measure? Explain. Chapter Skills Practice 645

20 Lesson.2 Skills Practice page Penny makes the following statement: Every rhombus is a parallelogram. Do you agree? Explain. 30. Sally plans to make the base of her sculpture in the shape of a rhombus. She cuts out four pieces of wood to create a mold for concrete. The pieces of wood are the following lengths: 5 inches, 5 inches, 3 inches, and 3 inches. Will the base of Sally s sculpture be a rhombus? Explain. 31. Ronald has a picture in the shape of a quadrilateral that he cut out of a magazine. How could Ronald use a ruler to prove that the picture is a parallelogram? 32. Three angles of a parallelogram have the following measures: 58, 122, and 58. What is the measure of the fourth angle? How do you know? 646 Chapter Skills Practice

21 Lesson.3 Skills Practice Name Date Kites and Trapezoids Properties of Kites and Trapezoids Vocabulary Write the term from the box that best completes each statement. base angles of a trapezoid midsegment biconditional statement isosceles trapezoid 1. The are either pair of angles of a trapezoid that share a base as a common side. 2. A(n) is a trapezoid with congruent non-parallel sides. 3. A(n) is a statement that contains if and only if. 4. The of a trapezoid is a segment formed by connecting the midpoints of the legs of the trapezoid. Problem Set Complete each statement for kite PRSQ. 1. PQ > QS and PR > SR P 2. /QPR > / Q T R 3. PT > 4. /PQT > / and /PRT > / S Chapter Skills Practice 64

22 Lesson.3 Skills Practice page 2 Complete each statement for trapezoid UVWX. _ 5. The bases are UV and WX. U V 6. The pairs of base angles are / and /, and / and /.. The legs are and. X W 8. The vertices are,,, and. Construct each quadrilateral using the given information. 9. Construct kite QRST with diagonals QS and RT intersecting at point M. Q S R T Q T M R S 648 Chapter Skills Practice

23 Lesson.3 Skills Practice page 3 Name Date 10. Construct kite LMNO with diagonals _ MO and LN intersecting at point A. M O L N 11. Construct kite UVWX with diagonals _ UW and XV intersecting at point K. U W X V Chapter Skills Practice 649

24 Lesson.3 Skills Practice page Construct trapezoid ABCD with AB as a base. A B 13. Construct trapezoid HGYU with HG as a base. H G 14. Construct trapezoid BNJI with BN as a base. B N 650 Chapter Skills Practice

25 Lesson.3 Skills Practice page 5 Name Date Use the given figure to answer each question. 15. The figure shown is a kite with DAB DCB. Which sides of the kite are congruent? A B D AB and CB are congruent. AD and CD are congruent. C 16. The figure shown is a kite with FG FE. Which of the kite s angles are congruent? F E G H 1. Given that IJLK is a kite, what kind of triangles are formed by diagonal IL? K I L J Chapter Skills Practice 651

26 Lesson.3 Skills Practice page Given that LMNO is a kite, what is the relationship between the triangles formed by diagonal _ MO? M N O L 19. Given that PQRS is a kite, which angles are congruent? Q P R S 20. Given that TUVW is a kite, which angles are congruent? T U W V 652 Chapter Skills Practice

27 Lesson.3 Skills Practice page Name Date Use the given figure to answer each question. 21. The figure shown is an isosceles trapezoid with AB CD. Which sides are congruent? A B C AC and BD are congruent. D 22. The figure shown is an isosceles trapezoid with EH FG. Which sides are parallel? E F H G 23. The figure shown is an isosceles trapezoid with IJ KL. What are the bases? I J L K 24. The figure shown is an isosceles trapezoid with _ MP NO. What are the pairs of base angles? M P N O Chapter Skills Practice 653

28 Lesson.3 Skills Practice page The figure shown is an isosceles trapezoid with PQ RS. Which sides are congruent? P Q R S 26. The figure shown is an isosceles trapezoid with LM i KN. What are the pairs of base angles? K L M N Create each proof. 2. Write a paragraph proof to prove /ABC > /ADC in kite ABCD. B A E C D You are given kite ABCD with diagonals BD and AC intersecting at point E. By definition of a kite, we know that AB > AD and BC > CD. By the Reflexive Property, you know that AC > AC. Therefore, n ABC > n ADC by the SSS Congruence Theorem. So, /ABC > /ADC by CPCTC. 654 Chapter Skills Practice

29 Lesson.3 Skills Practice page 9 Name Date 28. Write a two-column proof to prove HF > JF in kite GHIJ. I F H J G Chapter Skills Practice 655

30 Lesson.3 Skills Practice page Write a paragraph proof to prove _ WZ bisects /XWY and /XZY in kite WYZX. W X P Y Z 30. Write a paragraph proof to prove /U > /T in isosceles trapezoid RSTU. R S U X T 656 Chapter Skills Practice

31 Lesson.3 Skills Practice page 11 Name Date 31. Write a two-column proof to prove /K > /L in isosceles trapezoid JKLM. K L J N M Chapter Skills Practice 65

32 Lesson.3 Skills Practice page Write a two-column proof to prove that diagonals RT and SU in isosceles trapezoid RSTU are congruent if RU > ST. R S U T Construct each isosceles trapezoid using the given information. 33. Construct isosceles trapezoid ABCD if PS is the perimeter of the trapezoid. P S B C D P A S 658 Chapter Skills Practice

33 Lesson.3 Skills Practice page 13 Name Date 34. Construct isosceles trapezoid WXYZ if AD is the perimeter of the trapezoid. A D 35. Construct isosceles trapezoid EFGH if _ WZ is the perimeter of the trapezoid. W Z 36. Construct isosceles trapezoid PQRS if FJ is the perimeter of the trapezoid. F J Chapter Skills Practice 659

34 Lesson.3 Skills Practice page Construct isosceles trapezoid JKLM if _ TW is the perimeter of the trapezoid. T W 38. Construct isosceles trapezoid STUV if EH is the perimeter of the trapezoid. E H Use the given information to answer each question. 39. Alice created a kite out of two sticks and some fabric. The sticks were 10 inches and 15 inches long. She tied the sticks together so they were perpendicular and attached the fabric. When she measured the kite, she noticed that the distance from where the sticks meet to the top of the kite was 5 inches. What is the area of the kite Alice created? A 5 2 ( 1 2 ) (5)(5) 1 2 ( 1 2 ) (5)(10) A A 5 5 The area of the kite is 5 square inches. 660 Chapter Skills Practice

35 Lesson.3 Skills Practice page 15 Name Date 40. Simon connected a square and two congruent right triangles together to form an isosceles trapezoid. Draw a diagram to represent the isosceles trapezoid. 41. Magda told Sam that an isosceles trapezoid must also be a parallelogram because there is a pair of congruent sides in an isosceles trapezoid. Is Magda correct? Explain. 42. Sylvia drew what she thought was an isosceles trapezoid. She measured the base angles and determined that they measured 81, 9, 101 and 99. Could her drawing be an isosceles trapezoid? Explain. 43. Joanne constructed a kite with a perimeter of 38 centimeters so that the sum of the two shorter sides is 10 centimeters. What are the lengths of each of the two longer sides? 44. Ilyssa constructed a kite that has side lengths of 8 inches and 5 inches. What are the lengths of the other two sides? Explain. Chapter Skills Practice 661

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37 Lesson.4 Skills Practice Name Date Interior Angles of a Polygon Sum of the Interior Angle Measures of a Polygon Vocabulary Give an example of the term. 1. Draw an example of a polygon. Label the interior angles of the polygon. Problem Set Draw all possible diagonals from vertex A for each polygon. Then write the number of triangles formed by the diagonals A A 2 triangles 3. A 4. A Chapter Skills Practice 663

38 Lesson.4 Skills Practice page A A Use the triangles formed by diagonals to calculate the sum of the interior angle measures of each polygon.. Draw all of the diagonals that connect to vertex A. What is the sum of the interior angles of square ABCD? A B D C The diagonal divides the figure into two triangles. The sum of the interior angles of each triangle is 180º, so I multiplied 180º by 2 to determine the sum of the interior angles of the figure: 180º º The sum of the interior angles is 360º. 8. Draw all of the diagonals that connect to vertex E. What is the sum of the interior angles of figure EFGHI? E F I G H 664 Chapter Skills Practice

39 Lesson.4 Skills Practice page 3 Name Date 9. Draw all of the diagonals that connect to vertex J. What is the sum of the interior angles of figure JKMONL? J K L M N O 10. Draw all of the diagonals that connect to vertex P. What is the sum of the interior angles of the figure PQRSTUV? P Q V R U S T Chapter Skills Practice 665

40 Lesson.4 Skills Practice page Draw all of the diagonals that connect to vertex W. What is the sum of the interior angles of the figure WXYZABCD? W X D Y C Z B A 12. Draw all of the diagonals that connect to vertex H. What is the sum of the interior angles of the figure HIJKLMNOP? H I P J O N K M L 666 Chapter Skills Practice

41 Lesson.4 Skills Practice page 5 Name Date Calculate the sum of the interior angle measures of each polygon. 13. A polygon has 8 sides. The sum is equal to (n 2 2) 180º: (8 2 2) 180º º º The sum of the interior angles of the polygon is 1080º. 14. A polygon has 9 sides. 15. A polygon has 13 sides. 16. A polygon has 16 sides. 1. A polygon has 20 sides. 18. A polygon has 25 sides. Chapter Skills Practice 66

42 Lesson.4 Skills Practice page 6 The sum of the measures of the interior angles of a polygon is given. Determine the number of sides for each polygon (n 2 2) n n sides For each regular polygon, calculate the measure of each of its interior angles (n 2 2)180 (8 2 2)180 n _ (6) º The measure of each interior angle is 135º. 668 Chapter Skills Practice

43 Lesson.4 Skills Practice page Name Date Chapter Skills Practice 669

44 Lesson.4 Skills Practice page 8 Calculate the number of sides for each polygon. 31. The measure of each angle of a regular polygon is 108º. 108º 5 (n 2 2)180 n 108ºn 5 (n 2 2)(180º) 108ºn 5 180ºn 2 360º 2ºn 5 360º n 5 5 The regular polygon has 5 sides. 32. The measure of each angle of a regular polygon is 156º. 33. The measure of each angle of a regular polygon is 160º. 34. The measure of each angle of a regular polygon is 162º. 60 Chapter Skills Practice

45 Lesson.4 Skills Practice page 9 Name Date 35. The measure of each angle of a regular polygon is The measure of each angle of a regular polygon is Chapter Skills Practice 61

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47 Lesson.5 Skills Practice Name Date Exterior and Interior Angle Measurement Interactions Sum of the Exterior Angle Measures of a Polygon Vocabulary Identify the term in the diagram. 1. Identify the term that is illustrated by the arrow in the diagram below. Problem Set Extend each vertex of the polygon to create one exterior angle at each vertex Chapter Skills Practice 63

48 Lesson.5 Skills Practice page Calculate the sum of the measures of the exterior angles for each polygon.. pentagon hexagon 9. triangle 10. nonagon gon gon Given the measure of an interior angle of a polygon, calculate the measure of the adjacent exterior angle. 13. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 90º? Interior and exterior angles are supplementary. So subtract 90º, the measure of the interior angle, from 180º. 180º 2 90º 5 90º 64 Chapter Skills Practice

49 Lesson.5 Skills Practice page 3 Name Date 14. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 120º? 15. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 108º? 16. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 135º? 1. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 115º? 18. What is the measure of an exterior angle if it is adjacent to an interior angle of a polygon that measures 124º? Given the regular polygon, calculate the measure of each of its exterior angles. 19. What is the measure of each exterior angle of a square? _ º What is the measure of each exterior angle of a regular pentagon? Chapter Skills Practice 65

50 Lesson.5 Skills Practice page What is the measure of each exterior angle of a regular hexagon? 22. What is the measure of each exterior angle of a regular octagon? 23. What is the measure of each exterior angle of a regular decagon? 24. What is the measure of each exterior angle of a regular 12-gon? Calculate the number of sides of the regular polygon given the measure of each exterior angle _ 360 n n n sides Chapter Skills Practice

51 Lesson.6 Skills Practice Name Date Quadrilateral Family Categorizing Quadrilaterals Based on Their Properties Problem Set List all of the quadrilaterals that have the given characteristic. 1. all sides congruent 2. diagonals congruent square and rhombus 3. no parallel sides 4. diagonals bisect each other 5. two pairs of parallel sides 6. all angles congruent Identify all of the terms from the following list that apply to each figure: quadrilateral, parallelogram, rectangle, square, trapezoid, rhombus, kite.. 8. rhombus parallelogram quadrilateral Chapter Skills Practice 6

52 Lesson.6 Skills Practice page Name the type of quadrilateral that best describes each figure. Explain your answer Rectangle. The quadrilateral has two pairs of parallel sides and four right angles, but the four sides are not all congruent. 68 Chapter Skills Practice

53 Lesson.6 Skills Practice page 3 Name Date Chapter Skills Practice 69

54 Lesson.6 Skills Practice page 4 Draw the part of the Venn diagram that is described. 19. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals with four congruent sides. The other circle represents all types of quadrilaterals with four congruent angles. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. Four Congruent Sides Four Congruent Angles Rhombus Square Rectangle 20. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals with two pairs of congruent sides (adjacent or opposite). The other circle represents all types of quadrilaterals with at least one pair of parallel sides. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. 680 Chapter Skills Practice

55 Lesson.6 Skills Practice page 5 Name Date 21. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals with two pairs of parallel sides. The other circle represents all types of quadrilaterals with four congruent sides. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. 22. Suppose that part of a Venn diagram has two circles. One circle represents all types of quadrilaterals with four right angles. The other circle represents all types of quadrilaterals with two pairs of parallel sides. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. Chapter Skills Practice 681

56 Lesson.6 Skills Practice page Suppose that a Venn diagram has two circles. One circle represents all types of quadrilaterals with four congruent sides. The other circle represents all types of quadrilaterals with congruent diagonals. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. 24. Suppose that a Venn diagram has two circles. One circle represents all types of quadrilaterals with diagonals that bisect the vertex angles. The other circle represents all types of quadrilaterals with perpendicular diagonals. Draw this part of the Venn diagram and label it with the appropriate types of quadrilaterals. 682 Chapter Skills Practice

57 Lesson.6 Skills Practice page Name Date Tell whether the statement is true or false. If false, explain why. 25. A trapezoid is also a parallelogram. False. Parallelograms have two pairs of parallel sides. Trapezoids only have one pair. 26. A square is also a rhombus. 2. Diagonals of a rectangle are perpendicular. 28. A parallelogram has exactly one pair of opposite angles congruent. 29. A square has diagonals that are perpendicular and congruent. 30. All quadrilaterals have supplementary consecutive angles. List the steps to tell how you would construct the quadrilateral with the given information. 31. Rectangle HIJK given only diagonal HJ. 1. Duplicate HJ and bisect it to determine the midpoint. 2. Duplicate HJ again, labeling it IK, and bisecting it to determine the midpoint. 3. Draw segments HJ and IK so that their midpoints intersect. 4. Connect the endpoints of the segments to form rectangle HIJK. Chapter Skills Practice 683

58 Lesson.6 Skills Practice page Square ABCD given only diagonal AC. 33. Kite RTSU given only diagonal RS. 34. Rhombus MNOP given only diagonal MO. 35. Parallelogram JKLM given only diagonal JL. 36. Isosceles trapezoid BCDE given only diagonal BD. 684 Chapter Skills Practice