Geometry Semester 1 Final Review

Transcription

1 Geometry Semester 1 Final Review Short nswer 1. Name two lines in the figure. W T R. raw and label a pair of opposite rays and. 3. Name a plane that contains. W T R 4. Sketch a figure that shows two coplanar lines that do not intersect, but one of the lines is the intersection of two planes. 5. Find the length of is between and E. =, =, and E = 7. Find E. E 4x x 7. The map shows a linear section of Highway 35. Today, the Ybarras plan to drive the 360 miles from Springfield to Junction ity. They will stop for lunch in Roseburg, which is at the midpoint of the trip. If they have already traveled 55 miles this morning, how much farther must they travel before they stop for lunch?

2 S 55 mi X R 360 mi J 8. K is the midpoint of. and. Find JK, KL, and JL. 9. The tip of a pendulum at rest sits at point. uring an experiment, a physics student sets the pendulum in motion. The tip of the pendulum swings back and forth along part of a circular path from point to point. uring each swing the tip passes through point. Name all the angles in the diagram. O 10. m and m. Find m. I L J K 11. bisects, m, and m. Find m. 1. Tell whether and are only adjacent, adjacent and form a linear pair, or not adjacent. F G

3 13. Tell whether and are only adjacent, adjacent and form a linear pair, or not adjacent. F G 14. Tell whether and are only adjacent, adjacent and form a linear pair, or not adjacent. F G 15. Find the measure of the complement of, where m 16. Find the measure of the supplement of, where m 17. n angle measures degrees more than 3 times its complement. Find the measure of its complement. 18. billiard ball bounces off the sides of a rectangular billiards table in such a way that,, and and are complementary. If m, find m, m, and m

4 19. Name all pairs of vertical angles. J L M K N 0. Find the perimeter and area of the figure. 6 6x x The rectangles on a quilt are in. wide and 3 in. long. The perimeter of each rectangle is made by a pattern of red thread. If there are 30 rectangles in the quilt, how much red thread will be needed?. Find the circumference and area of the circle. Use 3.14 for, and round your answer to the nearest tenth. 4 ft 3. Find the coordinates of the midpoint of with endpoints (1, 6) and M(7, 5).

5 y 8 6 M x Find and EF. Then determine if. E y x 1 F Use the istance Formula and the Pythagorean Theorem to find the distance, to the nearest tenth, from T(4, ) to U(, 3). 6. There are four fruit trees in the corners of a square backyard with 30-ft sides. What is the distance between the apple tree and the plum tree P to the nearest tenth? N P F 7. Identify the transformation. Then use arrow notation to describe the transformation.

6 ' ' ' 8. Find the coordinates for the image of EFG after the translation (x, y) (x 6, y + ). raw the image. 7 y 7 7 x E F G 7 9. n animated film artist creates a simple scene by translating a kite against a still background. Write a rule for the translation of kite 1 to kite. 7 y 7 7 x Name three collinear points.

7 P R G N 31. Find the next item in the pattern, 3, 5, 7, 11, omplete the conjecture. The sum of two odd numbers is. 33. The table shows the population 65 years and over by age and sex according to the US ensus ureau, ensus 000 Summary file. Make a conjecture based on the data. Population 65 Years and Over by ge and Sex: 000 (numbers in thousands) 65 to 74 years 75 to 84 years 85 years and over Women 10,088 7,48 3,013 Men 8,303 4,879 1,7 34. Show that the conjecture is false by finding a counterexample. If, then. 35. Identify the hypothesis and conclusion of the conditional statement. If it is raining then it is cloudy. 36. Write a conditional statement from the statement. horse has 4 legs. 37. etermine if the conditional statement is true. If false, give a counterexample. If a figure has four sides, then it is a square. 38. Write the converse, inverse, and contrapositive of the conditional statement, If an animal is a bird, then it has two eyes. Find the truth value of each. 39. There is a myth that a duck s quack does not echo. group of scientists observed a duck in a special room, and they found that the quack does echo. Therefore, the myth is false. Is the conclusion a result of inductive or deductive reasoning? 40. etermine if the conjecture is valid by the Law of etachment. Given: If Tommy makes cookies tonight, then Tommy must have an oven. Tommy has an oven. onjecture: Tommy made cookies tonight. 41. etermine if the conjecture is valid by the Law of Syllogism.

8 Given: If you are in alifornia, then you are in the west coast. If you are in Los ngeles, then you are in alifornia. onjecture: If you are in Los ngeles, then you are in the west coast. 4. Write the conditional statement and converse within the biconditional. rectangle is a square if and only if all four sides of the rectangle have equal lengths. 43. etermine if the biconditional is true. If false, give a counterexample. figure is a square if and only if it is a rectangle. 44. Solve the equation. Write a justification for each step. Given equation [1] Simplify. [] Simplify. 45. gardener has 6 feet of fencing for a garden. To find the width of the rectangular garden, the gardener uses the formula, where P is the perimeter, l is the length, and w is the width of the rectangle. The gardener wants to fence a garden that is 8 feet long. How wide is the garden? Solve the equation for w, and justify each step. 46. Write a justification for each step. Given equation [1] Simplify. Subtraction Property of Equality Simplify. [] Simplify. Symmetric Property of Equality J M (6 x + 8)º ( x - 4)º K L m m = m + m [1] Substitution Property of Equality Simplify. Subtraction Property of Equality []

9 47. Identify the property that justifies the statement. and. So. 48. Write a justification for each step, given that. Symmetric Property of Equality E F G H Given information [1] Segment ddition Postulate [] Subtraction Property of Equality 49. Use the given plan to write a two-column proof. Given: m + m = 90, m + m = 90, m = m Prove: m = m Plan: Since both pairs of angle measures add to 90, use substitution to show that the sums of both pairs are equal. Since m = m, use substitution again to show that sums of the other pairs are equal. Use the Subtraction Property of Equality to conclude that m = m. omplete the proof. Proof: Statements Reasons 1. m + m = Given. [1]. Given 3. m + m = m + m 3. Substitution Property 4. m = m 4. Given 5. m + m = m + m 5. [] 6. m = m 6. [3] 50. Use the given paragraph proof to write a two-column proof.

10 1 3 Given: Prove: is a right angle. are complementary. Paragraph proof: Since is a right angle, m by the definition of a right angle. y the ngle ddition Postulate,. y substitution,. Since, by the definition of congruent angles. Using substitution,. Thus, by the definition of complementary angles, are complementary. omplete the proof. Two-column proof: Statements Reasons 1. is a right angle. 1. Given. m. efinition of a right angle [1] Substitution [] Substitution 7. are complementary. 7. efinition of complementary angles 51. Use the given two-column proof to write a paragraph proof. Given: 1 and are supplementary... Prove: 3 is a right angle. 1 3 Two-column proof: Statements Reasons 1. 1 and are supplementary. 1. Given... 1 and are right angles.. ongruent supplementary angles form right angles. 3. m 3. efinition of a right angle

11 4. 4. efinition of congruent angles 5. m 5. Substitution 6. 3 is a right angle. 6. efinition of a right angle omplete the proof. Paragraph proof: is a given statement. Since [1], 1 and are right angles. y the definition of a right angle, m. y the definition of congruent angles, []. Then, m by substitution. Therefore, 3 is a right angle by the definition of a right angle. 5. Identify a pair of parallel segments. H >> G E >> F >> >> 53. Give an example of corresponding angles Identify the transversal and classify the angle pair and.

12 n m l Find m. >> xº (3x - 70)º >> 56. Find m. R >> S (3x)º T U (4x 4)º >> V 57. Violin strings are parallel. Viewed from above, a violin bow in two different positions forms two transversals to the violin strings. Find x and y in the diagram.

13 (4x + y)º 100º (8x + y)º 60º 58. Use the onverse of the orresponding ngles Postulate and to show that. 1 l m 59. Use the information, and the theorems you have learned to show that. 1 l m 60. Write a two-column proof. Given: Prove: 3 1 l m omplete the proof.

14 Proof: Statements Reasons Given.. [1] Substitution (Steps 1 and ) [] 61. In a swimming pool, two lanes are represented by lines l and m. If a string of flags strung across the lanes is represented by transversal t, and x = 10, show that the lanes are parallel. (3x + 4)º l (4x 6)º t m 6. Write and solve an inequality for x. x Write a two-column proof. Given: Prove: t 1 m l omplete the proof. Proof: Statements 1. [1] 1. Given.. [] [3] Reasons

15 64. Use slopes to determine whether the lines are parallel, perpendicular, or neither. 65. Write the equation of the line with slope through the point (4, 7) in point-slope form. 66. Graph the line. 67. etermine whether the lines and are parallel, intersect, or coincide. 68. lassify by its angle measures, given m, m, and m. 5º 60º 75º 69. lassify by its side lengths is an isosceles triangle. is the longest side with length. = and =. Find. 8x +5 3x +9 4x + 4

16 71. jeweler creates triangular medallions by bending pieces of silver wire. Each medallion is an equilateral triangle. Each side of a triangle is 3 cm long. How many medallions can be made from a piece of wire that is 65 cm long? 3 cm 3 cm 3 cm 7. aphne folded a triangular sheet of paper into the shape shown. Find, given,, and m. E 4º 61º º 73. Find. L ( (4x + 7)º K (6x - 9)º 118º M N 74. Find and, given,, and.

17 N F E P M 75. Given: Identify all pairs of congruent corresponding parts. M O N 76. Given that and m = 3, find m. E 3º 77. Tom is wearing his favorite bow tie to the school dance. The bow tie is in the shape of two triangles. Given:,,, Prove: E omplete the proof. Proof:

18 Statements Reasons 1.,, 1. Given.. Given [1] [] 5. [3] 5. efinition of congruent triangles 78. Given the lengths marked on the figure and that bisects, use SSS to explain why. E 4 cm 4 cm 3 cm 3 cm 79. Show for. 6a - a + 7 4a - 16 omplete the proof... by the Reflexive Property of ongruence. So by [5]. 80. Given: P is the midpoint of and. Prove: T R P S Q

19 omplete the proof. Proof: Statements 1. P is the midpoint of and. 1. Given Reasons.,. [1] 3. [] 3. Vertical ngles Theorem [3] 81. etermine if you can use S to prove. Explain. E 8. etermine if you can use the HL ongruence Theorem to prove. If not, tell what else you need to know. P ^ ^ Q 83. pilot uses triangles to find the angle of elevation from the ground to her plane. How can she find m? 40 1 km 0 km O 0 km 1 km

20 84. Two Seyfert galaxies, W Tauri and M77, represented by points and, are equidistant from Earth, represented by point. What is m? Find m. P xº R (x + 15)º Q 86. Find. ) s + ) s 10 ) 87. Find the measures and.

21 6.4 X Given that bisects and, find. Y X Z W 89. are the perpendicular bisectors of. Find. 7.4 X 3.4 Y 4. O Z 90. Find the circumcenter of with vertices y x 3

22 91. are the angle bisectors of and, respectively., and m. Find m. O 40º 9. Point O is the centroid of,. Find. X Y O Z 93. Find the orthocenter of with vertices. 94. In, show that midsegment is parallel to and that. y (-4, ) K L x (4, -) (-4, -4) Given with,, and, find the length of midsegment. = 6 X Y = 5 3

23 96. anny and ana start hiking from the same base camp and head in opposite directions. anny walks 6 miles due west, then changes direction and walks for 5 miles to point. ana hikes 6 miles due east, then changes direction and walks for 5 miles to point S. Use the diagram to find which hiker is farther from the base camp. 5 mi 140º 6 mi base camp 6 mi R 130º 5 mi S 97. Write a two-column proof. Given:. Prove: F omplete the proof. Proof: Statements Given Reasons.. efinition of bisector 3. [1] 3. efinition of perpendicular [] [3] 98. Find the value of x. Express your answer in simplest radical form. x The size of a TV screen is given by the length of its diagonal. The screen aspect ratio is the ratio of its width to its height. The screen aspect ratio of a standard TV screen is 4:3. What are the width and height of a 7" TV screen?

24 height 7" width 100. Find the missing side length. Tell if the side lengths form a Pythagorean triple. Explain Tell if the measures 6, 14, and 13 can be side lengths of a triangle. If so, classify the triangle as acute, right, or obtuse. 10. Find the value of x. Express your answer in simplest radical form. 5 x x 103. n architect designs the front view of a house with a gable roof that has a triangle shape. The overhangs are 0.5 meter each from the exterior walls, and the width of the house is 16 meters. What should the side length l of the triangle formed with the overhangs be? Round your answer to the nearest meter. l l 16 m 0.5 m 0.5 m 104. Find the values of x and y. Express your answers in simplest radical form.

25 4 30º y 60º x 105. Tell whether the figure is a polygon. If it is a polygon, name it by the number of its sides Tell whether the polygon is regular or irregular. Tell whether it is concave or convex Find the measure of each interior angle of a regular 45-gon Find the measure of each exterior angle of a regular decagon The door on a spacecraft is formed with 6 straight panels that overlap to form a regular hexagon. What is the measure of?

26 X Y Z 110. The diagram shows the parallelogram-shaped component that attaches a car s rearview mirror to the car. In parallelogram RSTU, UR = 5, RX = 16, and = 4.4 o. Find ST, XT, and m RST. S T X R U 111. MNOP is a parallelogram. Find MP. M N 5x 3x+1 P O 11. Three vertices of parallelogram WXYZ are X(, 3), Y(0, 5), and Z(7, 7). Find the coordinates of vertex W Write a two-column proof. Given: F and F are parallelograms. Prove: F omplete the proof. Proof: Statements Reasons

27 1. F and F are parallelograms. 1. Given.. [1] Opposite sides in a parallelogram are parallel [] Substitution 114. Show that GHIJ is a parallelogram for x = 5 and y = 8. H 5x-10 I 3y 5y-16 G 7x-0 omplete the explanation. J Given [1] GJ = 7(5) 0 = [] Substitute and simplify. Given [3] [4] Substitute and simplify. Since HI = GJ and GH = JI, GHIJ is a parallelogram because [5] and. etermine if the quadrilateral must be a parallelogram. Justify your answer. L M K N 116. Show that quadrilateral EFG is a parallelogram. 10 y E.(3, 10) (-5,7) 5 F(8, 4) G (0,1) 5 5 x omplete the explanation.

28 and have the same slope, so [1]. Since E = FG, []. ecause [3], EFG is a parallelogram wooden frame has screws at,,, and so that the sides of it can be pressed to change the angles occurring at each vertex. and, even when the angles change. Why is the frame always a parallelogram? 118. n artist designs a rectangular quilt piece with different types of ribbon that go from the corner to the center of the quilt. The dimensions of the rectangle are inches and inches. Find. X 119. TRSU is a rhombus. Find. x + 5 R S 5x + T U 10. Show that all four sides of square are congruent and that.

29 5 y x etermine if the conclusion is valid. If not, tell what additional information is needed to make it valid. Given: onclusion: is a square. 1. Use the diagonals to determine whether a parallelogram with vertices is a rectangle, rhombus, or square. Give all the names that apply. 13. pillow is the shape of a kite. Heath wants to create a design connecting opposite corners from point to point, and from point to point. Find the amount of cording needing. One package of cording contains 5 inches of cord. How many packages does Heath need? 5 in. X 4 in. 10 in. 14. In kite PQRS, m and m. Find m.

30 Q P O R S 15. Given isosceles trapezoid with,, and. Find Y. Y 16. and. Find the value of x so that QRST is isosceles. R >> S Q >> T 17. Find RS. R >> S X Y Q >> 6

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