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NORTH THURSTON PUBLIC SCHOOLS END OF COURSE GEOMETRY PRACTICE TEST Name: _________________________________________________ Date: __________________________________________________ ** Video Tutorial Edition** Most questions below are linked to a video tutorial on the internet. To access a video, hold the control button while clicking on the link of the video you wish to watch. Day 1 1. Equilateral means all three sides are congruent, so one possible equation might be: 7.5๐ฅ = 6๐ฅ + 3 Determine the value of x if ฮABC is equilateral. B 7.5๐ฅ โ 6๐ฅ = 6๐ฅ โ 6๐ฅ + 3 1.5๐ฅ = 3 6x + 3 7.5x 10x โ 5 A 1.5๐ฅ 3 = 1.5 1.5 C ๐ฅ=2 Write your answer on the line. 2 What is the value of x? ______ _________ (Related Tutorial Video: Video 1) 2. Write the converse of the conditional statement. Determine if the converse is true or false. If it is false, find a counterexample. Original Statement: If P, then Q If you have a dog, then you are a pet owner. 0 A. 0 B. 0 C. 0 D. Converse Statement: If Q, then P If you are a pet owner, then you have a dog. True A dog owner owns a pet. True If you are a pet owner, then you have a dog. False, you could have a hamster. If you have a dog, then you are a pet owner. False, you could have a hamster. (Related Tutorial Video: Video 1) 3. 0 A. 0 B. 0 C. 0 D. Given aโb, determine which equation must be true. 1 2 3 4 a m ๏ 1 + m ๏ 5 = 180---No, < 1 โ < 5 5 6 m ๏ 3 + m ๏ 6 = 180---No < 3 โ < 6 b 7 8 m ๏ 2 + m ๏ 7 = 180---No < 2 โ < 7 m ๏ 4 + m ๏ 6 = 180---Yes, same side interior angles are supplementary! (Related Tutorial Videos: Video 1, Video 2, Video 3) < 1 = 60° (๐ฃ๐๐๐ก๐๐๐๐ <โฒ ๐ ) 70° Day 2 4. Determine measure of angle 2. Write your answer on the line. 2 70 + 60 + ๐ < 2 = 180 130 + ๐ < 2 = 180 1 60° 3 4 40° What is the measure of angle 2? ______50°_________degrees (Related Tutorial Videos: Video 1, Video 2) ๐ < 2 = 50° 5. ฮDEF has vertices D(4, 1), E(2, โ1), and F(โ2, โ1). Classify ฮDEF based on its sides. 0 A. equilateral 0 B. isosceles 0 C. scalene 0 D. right Work---see addendum at end of answer key! (Related Tutorial Videos: Video 1, Video 2) 6. Determine the equation of a line through the point (3, โ4) that is perpendicular to the line y = 3x + 7. 0 A. y = 3x โ 13 0 B. y = โ3x + 5 Perpendicular to y=3x + 7 means the slope of my line will be 1 the opposite reciprocal of 3, which is โ 3 1 ๏ญ xโ3 3 1 0 D. y = ๏ญ x โ 5 3 0 C. y = 1 ๐ฆ=โ ๐ฅ+๐ 3 (Related Tutorial Videos: Video 1, Video 2) 1 โ4 = โ (3) + ๐ 3 ๐ = โ3 Soโฆ.the equation is: 1 ๐ฆ=โ ๐ฅโ3 3 โ4 = โ1 + ๐ Day 3 7. Joe and Sara were standing on a pier sailing a toy sail boat. The boat was 6 feet from the base of the pier and the pier was 4 feet above the water. 4 6 ft tan ๐ = O 6 A ฦ 4 ft 4ft 4 ๐ก๐๐โ1 (๐ก๐๐๐) = ๐ก๐๐โ1 ( ) 6 ๐~33.7° B T Determine the angle of depression from the pier to the toy sail boat. Show your work using words, numbers and/or diagrams. What is the angle of depression from the pier to the sail boat?______33.7°________degrees (Related Tutorial Video: Video 1, Video 2, Video 3) 8. Joannaโs teacher said โThe diagonals of a square bisect each other.โ Joanna drew this figure and said โThe diagonals of this figure bisect each other, so it must be a square.โ The vertical diagonal is not bisected. Bisecting a line segment means the segment is separated into two congruent pieces. Joanna made an error in her mathematical argument. What is the error? 0 A. 0 B. 0 C. 0 D. There is no error. The figure Joanna drew is a square. In the figure Joanna drew, the diagonals do not bisect each other. Joanna used the converse of the teacherโs statement and the converse is false. Joannaโs teachers statement is false. The diagonals of a square do not bisect each other. (Related Tutorial Video: Video 1) 9. Determine the midpoint of JK , where J(โ1, 2) and K(6, 8). 1 , 5) 2 1 0 B. ( 2 , 5) 2 1 0 C. ( 3 , 3) 2 1 0 D. ( 2 , 3) 2 0 A. ( 3 (Related Tutorial Videos: Video 1, Video 2) Midpoint Formula: ( Midpoint: ( ๐1 +๐2 ๐1 +๐2 โ1+6 2+8 , 2 2 5 Midpoint: ( , 5) 2 Midpoint: (2 1 2, , 5) 2 ) , 2 ) Day 4 10. Look at the diagram. K M L L J N What theorem or postulate can you use to prove โ๐พ๐ฟ๐ โ โ๐๐ฟ๐ฝ? 0 A. Corresponding Parts of Congruent Triangles are Congruent ๐ < ๐พ๐ฟ๐ โ ๐ < ๐๐ฟ๐ฝ (Vertical angles are congruent) 0 B. Side-Angle-Side Congruence 0 C. Angle-Side-Angle Congruence 0 D. Side-Side-Side Congruence (Related Tutorial Video: Video 1, Video 2, Video 3) 11. Three vertices of a square have coordinates (3, 1), (4, -4) and (-1, -5). The diagonals of the square intersect at point Q. Since the diagonals of a square bisect each other, point Q will be the midpoint of either of the diagonals. Find the midpoint of the line segment joining points (3,1) & (โ1, โ5) ( 3 + (โ1) 1 + (โ5) , ) 2 2 (1, โ2) Determine the coordinates of point Q. You may use the blank grid to help determine the solution. Write your answer on the line. What are the coordinates of point Q? ( __1___ , __-2___ ) (Related Tutorial Videos: Video 1, Video 2 12. 3, 5, 7, and 11 are prime numbers. 4, 6, 8, 9, and 10 are composite numbers. Tiana makes the conjecture that prime numbers must be odd. Which statement is true? 0 A. This is an example of inductive reasoning and the conjecture is valid. 2 is a prime number which is even 0 B. This is an example of inductive reasoning and the conjecture is not valid. 0 C. This is an example of deductive reasoning and the conjecture is valid. 0 D. This is an example of deductive reasoning and the conjecture is not valid . (Related Tutorial Videos: Video 1, Video 2, Video 3) Day 5 13. One leg of a 45° โ 45° โ 90° triangle is 8 cm long. Determine the length of the hypotenuse. ๐ 0 A. 8โ3 ๐โ2 Soโฆ 8 8โ2 0 B. 8โ2 ๐ 0 C. 4โ2 8 0 D. 16 (Related Tutorial Videos: Video 1, Video 2) 14. Look at the conditional statement. Original Statement: If P, then Q โIf an angle measures 30°, then it is acuteโ Converse Statement: If Q, then P Which statement is the converse? 0 A. If an angle does not measure 30°, then it is not acute. 0 B. If an angle is not acute, then it does not measure 30°. 0 C. If an angle is acute, then it measures 30°. 0 D. If an angle measures 30°, then it is not acute. (Related Tutorial Video: See question 2) 15. Which equation represents the line through the points (-1, -2) and (2, 7)? 0 A. y = 3x + 1 โ๐ = 0 B. y โ 2 = 3(x โ 1) Slope: 0 C. y โ 7 = -3(x โ 2) ๐ฆ = ๐๐ฅ + ๐ 0 D. x โ 3y = 5 (Related Tutorial Videos: Video 1, Video 2, Video 3) โ๐ (7โ(โ2)) (2โ(โ1)) ๐ฆ = 3๐ฅ + ๐ 7 = 3(2) + ๐ 7= 6+๐ 9 = =3 3 1=๐ ๐ฆ = 3๐ฅ + 1 Day 6 16. Which statement is true about all parallelograms? 0 A. The diagonals bisect pairs of opposite angles. 0 B. The diagonals are congruent. 0 C. The diagonals bisect each other. 0 D. The diagonals are perpendicular (Related Tutorial Video: Video 1) 17. Quadrilateral ABCD is a rhombus and mโ BCE = 50°. A B The diagonals of a rhombus are perpendicular bisectors of each other, so ๐ < ๐ต๐ธ๐ถ = 90° ๐ < ๐ต๐ถ๐ธ + ๐ < ๐ต๐ธ๐ถ + ๐ < ๐ธ๐ต๐ถ = 180° 50° + 90° + ๐ < ๐ธ๐ต๐ถ = 180° E 140° + ๐ < ๐ธ๐ต๐ถ = 180° ๐ < ๐ธ๐ต๐ถ = 40° D C Determine the mโ EBC. Write your answer on the line. What is the mโ EBC? ____40_____ ° (Related Tutorial Videos: Video 1, Video 2 ) 18. Look at the conditional statement. โIf a figure is a pentagon, then it has five sidesโ Which statement is the inverse? 0 A. If a figure has five sides, then it is a pentagon. 0 B. If a figure is a pentagon, then it does not have five sides. 0 C. If a figure does not have five sides, then it is not a pentagon. 0 D. If a figure is not a pentagon, then it does not have five sides. (Related Tutorial Videos: Video 1, Video 2) Original Statement: If P, then Q Inverse Statement: If NOT P, then NOT Q Day 7 ฬ ฬ ฬ ฬ > ๐ฝ๐พ ฬ ฬ ฬ . 19. Triangle JKE is an obtuse isosceles triangle with mโ E = 10° and ๐พ๐ธ โJKE is isosceles, which means two of the angles are congruent. The fact that ฬ ฬ ฬ ฬ ๐พ๐ธ > ฬ ฬ ฬ ๐ฝ๐พ means ฬ ฬ ฬ ฬ that the angle opposite ๐พ๐ธ is larger than the angle opposite ฬ ฬ ฬ ๐ฝ๐พ (or angle J is larger than angle E). Since angle J is bigger than E and the triangle is isosceles, then angle K must measure 10° (a triangle canโt have two obtuse angles) giving us the equation: ๐ฝ + 10° + 10° = 180° What is the mโ J ? 0 A. 170° 0 B. 160° 0 C. 85° 0 D. 10° K E 10° J (Related Tutorial Videos: Video 1, Video 2) 20. A proof is shown. Fill in the blanks for steps 5 and 6 to complete the proof. Given: B is the midpoint of ฬ ฬ ฬ ฬ ๐ด๐ธ . B is the midpoint of ฬ ฬ ฬ ฬ ๐ถ๐ท . A C B Prove: โ๐ด๐ต๐ท โ โ๐ธ๐ต๐ถ. D E Statements Reasons 1. B is the midpoint of ฬ ฬ ฬ ฬ ๐ด๐ธ . 1. Given 2. ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ๐ด๐ต โ ๐ต๐ธ 2. Definition of midpoint 3. B is the midpoint of ฬ ฬ ฬ ฬ ๐ถ๐ท . 3. Given 4. ฬ ฬ ฬ ฬ ๐ถ๐ต โ ฬ ฬ ฬ ฬ ๐ต๐ท 4. Definition of midpoint 5. m<ABD โ m<CBE 5. Vertical Angles Theorem 6. โ๐ด๐ต๐ท โ โ๐ธ๐ต๐ถ 6. Side-Angle-Side Congruency 21. Which ordered pair is the midpoint of the line segment with endpoints (2,-5) and (-6, 4)? Midpoint Formula: ( 0 A. (-4, -1) ๐1 +๐2 ๐1 +๐2 , 2 1 0 B. (โ4, โ ) 2 1 0 C. (โ2, โ ) 2 0 D. (-2, -1) (Related Tutorial Video: See question 9) Midpoint: ( 2 + (โ6) โ5 + 4 2 , 1 Midpoint: (โ2, โ ) 2 2 ) 2 ) The diagonal of a square creates an isosceles triangle (or a 45°-45°-90° triangle) with the following side ratios: Day 8 ๐ฅโ2 22. The diagonal of a square is 7 cm. X X 7 cm Since the hypotenuse of the triangle (or the diagonal of the square) is 7, we can write the following equation: Determine the length of one side of the square. 0 A. 3 1 2 0 B. 7โ2 0 C. 0 D. ๐ฅโ2 = 7 If we divide each side of the equation by โ2, we get: 7 ๐ฅ= โ2 Mathematical conventions do not allow us to have a radical (square root sign) in the denominator of a fraction. We 7โ3 3 rationalize the fraction by multiplying by 7โ2 7 โ2 7โ2 โ2 โ2 โ4 like: ( ) ( ) = 2 = โ2 โ2 , which looks 7โ2 2 (Related Tutorial Videos: Video 1, Video 2, Video 3) 23. Which statement is true? 0 A. A postulate is accepted as true without proof. 0 B. A theorem is accepted as true without proof. 0 C. Definitions can never be used as reasons in a proof. 0 D. Theorems can never be used as reasons in a proof. (Related Tutorial Video: Video 1) 24. Two unique planes intersect. Which geometric term describes the intersection? 0 A. line 0 B. plane 0 C. point 0 D. segment (Related Tutorial Video: Video 1 A plane is like a never-ending sheet of paper. To visualize two planes intersecting, hold the edges of two sheets of paper together. They intersect in an infinite arrangement of points in a straight line Opposite angles in a parallelogram are congruent. Therefore, 6๐ โ 7 = 4๐ + 20 Day 9 2๐ = 27 25. Look at the diagram. ๐ = 13.5 (6๐ โ 7)° ๐° Angle Measure: (6)(13.5)-7=74° Consecutive angles in a parallelogram are supplementary (add up to 180°). Therefore, (4๐ + 20)° 74 + ๐ = 180° Determine the values for a and b that would make the quadrilateral a parallelogram. ๐ = 106° 0 A. a = 13.5, b = 166.5 0 B. a = 16.7, b = 93.2 0 C. a = 13.5, b = 106 0 D. a = 16.7, b = 86.8 (Related Tutorial Video: See question 16) Work---see addendum at end of answer key! 26. Look at the triangle. 18 13 5 y Determine the length of y. Express your answer in simplified radical form. Write your answer on the line. What is the length of y? 6โ5 (Related Tutorial Video: Video 1) 27. Lines l, m, and n lie in the same plane. Line m is perpendicular to line l. Line n is perpendicular to line l. Which statement is true? 0 A. Line m and line n are perpendicular. 0 B. Line m and line n are parallel. 0 C. Line m and line n will intersect. 0 D. Line m and line n are skew Line m Line n Line l Day 10 28. While walking around Seattle, Mary climbed several steep streets. One of the steepest streets, Roy Street has a slope angle of 11.9° according to the tour guide. After walking 100 feet up the hill, she wanted to determine how high she had climbed. 11.9° 100 feet Height climbed Horizontal distance Use a trigonometric ratio (sine, cosine, tangent) to determine how high Mary climbed. Be sure to write the equation and show the steps you used to solve the equation. Round your answer to the nearest foot. sin 11.9° = โ๐๐๐โ๐ก 100 โ๐๐๐โ๐ก (100)(sin 11.9) = ( ) (100) 100 20.6 โ โ๐๐๐โ๐ก How high did Mary climb?โ 20.6 ๐๐๐๐ก (Related Tutorial Videos: See question 7, Video 1) 29. Determine how many miles a person will run during a 5-kilometer race. Write your answer on the line. 1 km โ 0.62 mi ( 5 ๐๐๐๐๐๐๐ก๐๐๐ . 62 ๐๐๐๐๐ )( ) = 3.1 ๐๐๐๐๐ 1 1 ๐๐๐๐๐๐๐ก๐๐ How many miles will a person run during a 5-kilometer race?3.1 ๐๐๐๐๐ (Related Tutorial Videos: Video 1, Video 2) 30. Which statement is an example of inductive reasoning? Inductive Reasoning: Based on observation and predictions. 0 A. M is a midpoint of AB. Therefore AM = MB. 0 B. Squares have equal sides. This figure has equal sides, therefore this figure is a square. 0 C. The base angles of an isosceles triangle are equal. The base angles of this triangle are equal. Therefore, this triangle is isosceles. 0 D. Triangular numbers have a pattern. 1, 3, 6, 10, and 15 are triangular numbers. Therefore, the next triangular number is 21. (Related Tutorial Videos: See question 12) Day 11 31. The rhombus QRST is made of two congruent triangles. Given m ๏ QRS = 34° determine the measure of ๏ S. 0 A. 0 B. 0 C. 0 D. R Q What is the measure of ๏ S? 56° 73° 112° 146° S T Rhombus means 4 equal sides, so triangle TSR mush be isosceles. Diagonal TR bisects angle QRS, so angle TRS measures 17°. ฬ ฬ ฬ ฬ โ ฬ ๐๐ ฬ ฬ ฬ , < ๐๐ ๐ โ < ๐ ๐๐ (both angles are Since ๐ ๐ 17°) 17° + 17° + ๐ = 180 ๐ = 146° (Related Tutorial Videos: Video 1) 32. Determine whether the conjecture is true or false. Give a counterexample for a false conjecture. Given: points A, B, C, and D Conjecture: A, B, C, and D are coplanar True or false? False Coplanar means all fitting on the same plane (imagine an endless sheet of paper). Any 4 points do not necessarily have to be in the same plane. Consider the 4 points that make up the vertices of a cube----these 4 points do not all lie in the same plane. (Related Tutorial Video: Video 1) 33. Identify the congruent triangles in the diagram and write a congruence statement. M L S K 0 A. 0 B. 0 C. 0 D. ฮ LSM ๏ ฮ LSK ๏ ฮ LKS ๏ ฮ LSK ๏ ฮ KSN ฮ MSN ฮ MSN ฮ NSM (Related Tutorial Video: Video 1) N Although Choices B, C, and D all name the same two triangles, the order in choices C and D are wrong. Corresponding sides must match ฬ ฬ ฬ โ ฬ ฬ ฬ ฬ up in naming congruent triangles (๐ฟ๐ ๐๐ so the LS in naming โLSK must be in the same position as the MS in naming โMSN) Day 12 34. Determine cos I in ฮGHI. G 75 21 0 A. 0 B. 0 C. 0 D. H 21 72 21 75 75 72 72 75 I 72 cos = ๐๐๐๐๐๐๐๐ก โ๐ฆ๐๐๐ก๐๐๐ข๐ ๐ cos ๐ผ = 72 75 (Related Tutorial Video: Video 1) 35. Write the contrapositive of the conditional statement. Determine if the contrapositive is true or false. If it is false, find a counterexample. Original Statement: If P, then Q โTwo angles measuring 180 degrees are supplementaryโ 0 A. 0 B. 0 C. 0 D. Contrapositive: If not Q, then not P Two angles not measuring 180 degrees are supplementary. True More than two angles measuring 180 degrees are non-supplementary. True Non-supplementary angles are not two angles measuring 180 degrees. True Non-supplementary angles are two angles measuring 180 degrees. False; supplementary angles must measure 180 degree. (Related Tutorial Videos: See question 18) 36. Complete this chart. Figure Triangular Pyramid Triangular Prism Square Pyramid Cube Hexagonal Pyramid Hexagonal Prism # of edges e 6 9 8 12 12 18 # of faces f 4 5 5 6 7 8 (Related Tutorial Videos: Video 1, Pyramid Definition, Prism Definition) # of vertices v 4 6 5 8 7 12 f+v 8 9 10 14 14 20 Day 13 37. Determine which statement is a property of all rectangles. 0 A. 0 B. 0 C. 0 D. Four congruent sides. Diagonals bisect the angles. Diagonals are perpendicular. Four right angles. (Related Tutorial Video: Video 1, Video 2) 38. The figure is a rectangular prism with dimensions 12 inches long, 5 inches wide and 7 inches tall. Determine the length of BI . B Work---see addendum at end of answer key! C A D 7 G H 5 F Write your answer on the line. I 12 What is the length of BI ?__________________ (Related Tutorial Video: Video 1) 39. Given B(โ4, โ6), determine which reflection would result in Bโ(6, 4). 0 A. 0 B. 0 C. 0 D. Reflected over the x-axis. Reflected over the y-axis. Reflected over the line y = โx. Reflected over the line y = x. (Related Tutorial Video: Video 1) Day 14 The ratio for a 30°-60°-90° triangle are: 40. Determine the exact length of x in ฮHJK. K 0 A. 5 2 0 B. 5 0 C. 10 0 D. 15 60° x 3 2๐ ๐ 60° 30° 5 ๐โ3 J (Related Tutorial Videos: Video 1, Video 2) 30° H ฬ ฬ ฬ ๐ค๐๐๐ ๐๐ ๐ก๐ค๐๐๐ ๐๐ ๐๐๐ ๐๐ ๐พ๐ป ฬ ฬ ฬ ฬ ๐พ๐ฝ ฬ ฬ ฬ ๐พ๐ฝ = (2)(5) = 10 41. If you are going 50 miles per hour, determine how many feet per second you are traveling. Write your answer on the line. 50 ๐๐๐๐๐ 5280 ๐๐๐๐ก 1 โ๐๐ข๐ 1 ๐๐๐๐ข๐ก๐ 264000 ๐๐๐๐ก ๐๐ก ( )( )( )( )= โ 73.3 1 โ๐๐ข๐ 1 ๐๐๐๐ 60 ๐๐๐๐ข๐ก๐๐ 60 ๐ ๐๐๐๐๐๐ 3600 ๐ ๐๐๐๐๐๐ ๐ ๐๐ How many feet per second are you traveling? โ 73.3 ๐๐ก ๐ ๐๐ (Related Tutorial Videos: See question 29) 42. The ratio of a pair of corresponding sides in two similar triangles is 5:3. The area of the smaller triangle is 108 in2. What is the area of the larger triangle? 0 A. 300 in2 If the Ratio of Similarity is a:b, then the Ratio of Area is a2 : b2 0 B. 180 in2 The Ratio of Area for the two triangles is: 52 : 32 OR 25:9 2 0 C. 64.8 in 0 D. 38.9 in2 (Related Tutorial Videos: Video 1 Write a proportion to solve: 25 9 = ๐ฅ 108 Cross-products gives us the equation: 9๐ฅ = 2700 ๐ฅ = 300 Day 15 43. Jessie is working on the roof of her house. She has measured the angle of the roof and the length of the roof. Determine the width of the house, w. 15 ft. 15 ft. 35° 0 A. 18.31 ft. w 0 B. 24.57 ft. 0 C. 12.29 ft. 0 D. 26.15 ft. (Related Tutorial Video: See question 28, Video 1, Video 2) Begin by dropping an altitude from the peak of the roof to form two congruent right triangles (the base will be half the length of ๐ค). Since the triangle is a right triangle, we can use trigonometry to solve. ๐ค ( ) cos 35° = 2 15 ๐ค ( ) (15)(cos 35°) = 2 (15) 15 ๐ค 12.287 โฆ โ ( ) 2 24.57 โ ๐ค 15 ft 35° ๐ค 2 44. The segment bisector is the midpoint. Write the inverse. The point that does not bisect a segment is not the midpoint Determine if the statement is true or false. ____________TRUE_________________________ If it is false, write a counterexample. (Related Tutorial Video: See question 18) 45. ฮRST has vertices R(3, 3), S(6, โ2), and T(0, โ2). Classify ฮRST based on its sides. 0 A. isosceles 0 B. scalene 0 C. equilateral 0 D. right (Related Tutorial Videos: See question 5) ฬ ฬ ฬ ฬ = ๐ ๐๐๐๐๐ (by counting) ๐ป๐บ ฬ ฬ ฬ ฬ = โ(3 โ 6)2 + (3 โ (โ2))2 ๐ ๐ R ฬ ฬ ฬ ฬ ๐ ๐ = โ(โ3)2 + (5)2 ฬ ฬ ฬ ฬ ๐น๐บ = โ๐๐ T S ฬ ฬ ฬ ฬ = โ(0 โ 3)2 + (โ2 โ 3)2 ๐ ๐ ฬ ฬ ฬ ฬ ๐ ๐ = โ(โ3)2 + (โ5)2 ฬ ฬ ฬ ฬ = โ๐๐ ๐น๐ป โRTS is Isosceles Day 16 46. Look at the given information for quadrilateral ABCD. ฬ ฬ ฬ ฬ ฬ ฬ ฬ ฬ ๐ด๐ต ||๐ถ๐ท ฬ ฬ ฬ ฬ ๐ด๐ท โ ฬ ฬ ฬ ฬ ๐ต๐ถ ฬ ฬ ฬ ฬ ๐ด๐ท is not parallel to ฬ ฬ ฬ ฬ ๐ต๐ถ ๏ท ๏ท Draw and label a shape that satisfies all of the given information. Determine the most specific name for the shape. A B > D > C What is the most specific name for quadrilateral ABCD? Isosceles Trapezoid (Related Tutorial Video: Video 1) 29๐๐3 literally means 29(๐๐)(๐๐)(๐๐) 47. Martina has a calculator box that has a volume of 29 cubic inches. The conversion looks like: 1 inch = 2.54 centimeters ( 29(๐๐)(๐๐)(๐๐) 2.54 ๐๐ 2.54 ๐๐ 2.54 ๐๐ )( )( )( ) 1 1 ๐๐ 1 ๐๐ 1 ๐๐ Determine the volume of the calculator box to the nearest cubic centimeter. โ 475.22๐๐3 Write your answer on the line. What is the volume of the calculator box to the nearest cubic centimeter? โ ๐๐๐. ๐๐ ๐๐๐ (Related Tutorial Video: Video 1) 48. Determine the image of Y(โ4, 7) under the translation of (x, y) ๏ฎ (x + 3, y โ 5). 0 A. Yโ(โ1, 2) 0 B. Yโ(โ1, 12) 0 C. Yโ(โ7, 2) 0 D. Yโ(โ7, 12) (Related Tutorial Video: Video 1) (x, y) ๏ฎ (x + 3, y โ 5) (โ4, 7) โ (โ4 + 3, 7 โ 5) (โ1, 2) G Day 17 18 E 49. Look at the figure. 8 F 10 12 D H What additional information do you need to show the triangles below are similar using the Side-Angle-Side Similarity Theorem? Since we have an angle already (the vertical angle) and we want Side-AngleSide congruency, choice B and C will be eliminated as possibilities as they 0 A. FH = 12 give us additional angles. Choice C gives us a side with the correct ratio 10 12 2 0 B. โ E โ โ H = = , however will not result in Side-Angle-Side similarity. Choice A 15 0 C. GH = 15 18 3 will give us the correct ration still 12 18 = 8 12 = 2 3 0 D. โ D โ โ G (Related Tutorial Videos: Video 1, Video 2 50. Find sin C as a decimal rounded to the nearest hundredth. sin = ๐๐๐๐ ๐๐๐๐๐ ๐๐ก๐ ๐ป๐ฆ๐๐๐ก๐๐๐ข๐ ๐ C sin ๐ถ = 0 A. 0.49 16.9 8.3 14.7 sinโ1 (sin ๐ถ) = sinโ1 ( ) 16.9 0 B. 60.44° 0 C. 0.87° 0 D. 0.87 A 14.7 16.9 B 14.7 ๐ด๐๐๐๐ ๐ถ โ 60.44° (Related Tutorial Video: Video 1) 51. Two similar figures have a ratio of volumes of 64:27. What is the ratio of similarity? Write your answer on the line. What is the ratio of similarity? (Related Tutorial Video: Video 1) ๐ ๐๐ก๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐ก๐ฆ ๐๐ ๐ ๐๐ก๐๐ ๐๐ ๐๐๐๐ข๐๐ ๐๐ ๐ ๐ ๐ Note: Taking the cube root of the Ratio of Volume will give you the Ratio of Similarity. ๐ ๐3 ๐3 3 ๐ ๐๐ก๐๐ ๐๐ ๐๐๐๐๐๐๐๐๐ก๐ฆ: โ( 64 4 )= 27 3 Day 18 52. Michael is 5 1 feet tall. Michael measures his shadow as 8 feet long. A tree in his backyard has a shadow that is 25 yards long. How tall is the tree? Set up a proportion: 2 ๐๐๐โ๐๐๐โฒ ๐ ๐ป๐๐๐โ๐ก ๐๐๐โ๐๐๐๐ ๐โ๐๐๐๐ค = ๐๐๐๐ ๐ป๐๐๐โ๐ก ๐๐๐๐ ๐โ๐๐๐๐ค 5.5 ๐๐๐๐ ๐ป๐๐๐โ๐ก = 8 25 5 ½ ft 8 ft 0 A. 51.56 yards Using cross products we find the treeโs height= 17.1875 ๐ฆ๐๐๐๐ . Since 17.1875 yards isnโt a choice, we have to convert the yards to feet by multiplying by the 3 ๐๐๐๐ก conversion factor of , which gives us โ 51.56 ๐๐๐๐ก 0 B. 17.19 feet 0 C. 51.56 feet 1 ๐ฆ๐๐๐ 0 D. 36.36 yards (Related Tutorial Video: Video 1) 25 yd 53. Look at the pair of triangles. B A D C Which statement is true? 0 A. The triangles are congruent. 0 B. The triangles are similar but not congruent. 0 C. The triangles are not similar or congruent. 0 D. There is not enough information to determine similarity or congruence. (Related tutorial Videos: See question 10) 54. Steven built a box for his vegetable garden in the shape of a rectangular prism. The volume of the vegetable garden was 24 cubic feet. He built another garden box that was two times longer and two times higher. He thinks the volume will be twice as much. Explain why Steven is not correct. Volume=(2๐ฅ)(๐ฆ)(2๐ง) = 4๐ฅ๐ฆ๐ง Volume = (๐ฅ)(๐ฆ)(๐ง) z 2z y y x 2x 54 continued on next page By doubling 2 dimensions, Steve has actually increased the Volume by a factor of 4 54 cont. If Steven wants his second garden box to have twice the volume, what should he do instead? If Steve wants to double the volume he should consider only doubling one dimensionโฆso ๏ท Make the box twice as long OR ๏ท Make the box twice as high OR ๏ท Make the box twice as wide OR ๏ท 3 Increase all three dimensions by a factor of โ2 (Related Tutorial Video: See question 51) Day 19 tan ๐บ = E 55. To the nearest degree, what is mโ G? 12.1 tanโ1 (tan ๐บ) = tanโ1 ( ) 7.8 0 A. 2° ๐บ โ 57.19° 0 B. 57° 12.1 0 C. 50° 12.1 7.8 0 D. 33° F G 7.8 (Related Tutorial Video: Video 1, See question 7) 56. JKLM is an isosceles trapezoid with J(0, โ1), K(โ2, 3) and M(6, โ1). Determine the coordinates of L. 0 A. 0 B. 0 C. 0 D. L (6, 1) L (9, 4) L (2, 3) L (8, -3) Work---see addendum at end of answer key! (Related Tutorial Video: Video 1) 57. Given aโb, determine which relationship must be true. 0 A. 0 B. 0 C. 0 D. ๏ 1 and ๏ 2 and ๏ 4 and ๏ 3 and ๏ 3 are congruent. ๏ 8 are supplementary. ๏ 5 are similar. ๏ 7 are complementary. (Related Tutorial Video: See question 3) a b 3 7 5 6 8 1 2 4 Day 20 58. Determine which triangle is similar to ฮDEF. D 13 5 E 12 3 0 A. 5 0 B. F 12 The triangle from C has the same side length ratios as the triangle above: 10 5 12 13 1 = = ๐๐๐ ๐๐ ๐กโ๐๐ ๐ ๐๐๐ก๐๐๐ ๐ ๐๐๐๐๐๐๐ฆ ๐๐๐ค๐ ๐ก๐ 15 36 39 3 17 18 36 0 C. 15 39 0 D. 5 3 4 (Related Tutorial Videos: Video 1, Video 2) B 59. Name all segments skew to C BC . A D 0 A. FI , AD, FA, DI 0 B. FG, GH , HI , FI 0 C. CD, AB, BG, CH 0 D. GF , HI , DI , AF H G (Related Tutorial Video: Video 1) F I 60. Determine which theorem or postulate can be used to prove that these two triangles are similar. 8 63° 4 27° 6 0 A. 0 B. 0 C. 0 D. Since both triangles have two angles which are given, we can use the triangle sum theorem (which says the sum of the angles in any triangle must add up to 180°) to find the missing angles of both triangles. By doing this we find that both triangles have angle measures of 27°, 63°, ๐๐๐ 90° AA SAS SSA SSS (Related Tutorial Videos: See question 58) Day 21 61. Cody is standing 14 feet from the base of the tree. The top of the tree makes a 60° angle with the ground at the point where he is standing. ๐๐๐๐ ๐ป๐๐๐โ๐ก tan 60° = ( ) 14 Tree 14(tan 60°) = ( 60° ๐๐๐๐ ๐ป๐๐๐โ๐ก ) (14) 14 24.25 โ ๐๐๐๐ ๐ป๐๐๐โ๐ก 14 ft 60° 14 ft Determine the height of the tree. Round your answer to two decimal places. Write your answer on the line. What is the height of the tree? โ ๐๐. ๐๐feet (Related Tutorial Videos: See question 28, Video 1) 62. Given aโb and m ๏ 3 = 5x + 10 and m ๏ 5 = 3x + 10, determine the value of x. 0 A. 0 B. 0 C. 0 D. 110 70 20 2.5 b (Related Tutorial Videos: See question 3) 1 2 3 4 a 7 5 6 8 Angle 1 and Angle 5 are co-interior (or same side interior) angles on parallel lines and are therefore supplementary. We can write the equation: (5๐ฅ + 10) + (3๐ฅ + 10) = 180° 8๐ฅ + 20 = 180 8๐ฅ = 160 ๐ฅ = 20 ฬ = 72 and ๐ด๐ท ฬ ฬ ฬ ฬ is a diameter. Determine m ๏ BCD. 63. In circle C, ๐๐ด๐ต Write your answer on the line. ฬ = 72, then the measure of Since ๐๐ด๐ต angle ACB is 72°. Since ฬ ฬ ฬ ฬ ๐ด๐ท is a diameter and measures 180°, <BCD = 180-72 = 108° A C D B What is the m ๏ BCD? 108° degrees (Related Tutorial Videos: Video 1) Day 22 64. Determine which set of measures could represent the sides of a right triangle. 0 A. 0 B. 0 C. 0 D. 2, 3, 4 7, 11, 14 8, 10, 12 9, 12, 15 The only triple that satisifies the Pythagorean Theorem (๐2 + ๐ 2 = ๐ 2 ) is the side lengths of 9, 12, and 15 because 92 + 122 = 152 (81 + 144 = 225) (Related Tutorial Video: Video 1) 65. Determine the exact value of x in ฮLMN. N Work---see addendum at end of answer key! 0 A. 25 25 2 0 C. 25 3 0 B. 50 x 0 D. 100 (Related Tutorial Video: See question 40) L 60° M 66. ฮRST has vertices R(-2 5), S(1, 1), and T(-6, 2). Classify ฮRST based on its sides. 0 A. 0 B. 0 C. 0 D. isosceles scalene right isosceles right (Related Tutorial Video: See question 5) Day 23 67. In the diagram: A Given: AB ๏ EB ๏D ๏ ๏C Prove: ๏ABD ๏ ๏EBC C B D E Proof: 1. 2. 3. 4. Statement ฬ ฬ ฬ ฬ ๐ด๐ต โ ฬ ฬ ฬ ฬ ๐ธ๐ต < ๐ท โ < ๐ถ < ๐ด๐ต๐ท โ < ๐ถ๐ต๐ธ โ๐ด๐ต๐ท โ โ๐ธ๐ต๐ถ 1. 2. 3. 4. Justification Given Given Vertical Angles are Congruent AAS Congruency (Related Tutorial Video: See question 10) The product of segments of an intersecting chords are equal, so we know: 68. Determine the value of y. (3)(9) = (6)(๐ฆ) 3 0 A. 0 B. 0 C. 0 D. 18 12 6 4.5 27 = 6๐ฆ y 6 4.5 = ๐ฆ 9 (Related Tutorial Video: Video 1, Video 2) G 69. For rhombus GHJK, determine m ๏ 1. 0 A. 0 B. 0 C. 0 D. 45° 60° 90° 120° (Related Tutorial Video: See question 17) H 1 K J Day 24 70. Given quadrilateral XYWZ, determine whether WX and YZ are parallel, perpendicular or neither. W(0, โ3), X(โ1, 5), Y( 2, 5) Z(โ1, 2) Show your work using words, numbers and/or diagrams. ๐๐๐๐๐ ฬ ฬ ฬ ฬ ฬ ๐๐ = 5 โ (โ3) 8 = = โ8 โ1 โ 0 โ1 ๐๐๐๐๐ ฬ ฬ ฬ ฬ ๐๐ = 2โ5 โ3 = =1 โ1 โ 2 โ3 Are There is neither a perpendicular or parallel relationship ฬ ฬ ฬ ฬ ฬ ๐๐๐ ฬ ฬ ฬ ฬ between the sides ๐๐ ๐๐ WX and YZ parallel, perpendicular or neither? Neither (Related Tutorial Video: See question 6) 71. Determine the value of x and y so that QRST will be a parallelogram. Q 0 A. 0 B. 0 C. 0 D. R 24° x = 6, y = 42 x = 6, y = 22 x = 20, y = 42 x = 20, y = 22 (y-10)° 32° 4x° T (Related Tutorial Video: See question 16) If QRST is a parallelogram, then alternate interior angles should be equal. That means: 4๐ฅ = 24 ๐ฅ=6 And ๐ฆ โ 10 = 32 ๐ฆ = 42 S 72. Two angles measuring 90° are complementary. Write the inverse. Two angles whose sum is not 90° are not complementary Determine if the statement is true or false. ______TRUE_______________________________ If it is false, write a counterexample. (Related Tutorial Video: See question 18) Day 25 73. Which statement is an example of deductive reasoning? 0 A. 0 B. 0 C. 0 D. This bird is white. Swans are white. Therefore, this bird is a swan. Dogs are mammals. Mammals breathe oxygen. Therefore dogs breathe oxygen. This rock is not heavy. Lava rocks are not heavy. Therefore this rock is lava rock. When the sidewalks get icy, they get slippery. The sidewalks are slippery. Therefore the sidewalks are icy. (Related Tutorial Video: See question 12) AF ๏ DE, AB ๏ FC and AB โ FC , determine which theorem or postulate can be used to prove ๏ABE ๏ ๏FCD . 74. If B 0 A. 0 B. 0 C. 0 D. <BAC โ <CFE because they are corresponding angles from parallel lines, so we have a Side-AngleSide C AAS ASA SAS SSS A F (Related Tutorial Video: See question 10) E D 75. Identify which of the following is a property of a parallelogram. 0 A. 0 B. 0 C. 0 D. The diagonals are congruent. The diagonals bisect the angles. The diagonals are perpendicular. The diagonals bisect each other. (Related Tutorial Video: See question 16) Addendum to Answer Key 5. ฮDEF has vertices D(4, 1), E(2, โ1), and F(โ2, โ1). Classify ฮDEF based on its sides. ฬ ฬ ฬ ฬ ๐น๐ธ = 4 ๐ข๐๐๐ก๐ ๐๐๐๐ (๐๐๐ข๐๐ก๐๐) ฬ ฬ ฬ ฬ = โ(4 โ (โ2))2 + (1 โ (โ1))2 ๐น๐ท ฬ ฬ ฬ ฬ = โ62 + 22 ๐น๐ท D ฬ ฬ ฬ ฬ = โ40 ๐น๐ท F 2 ฬ ฬ ฬ ฬ ๐ธ๐ท = โ(4 โ 2)2 + (1 โ (โ1)) E ฬ ฬ ฬ ฬ ๐ธ๐ท = โ22 + 22 ฬ ฬ ฬ ฬ ๐ธ๐ท = โ8 ***All sides are a different length---this is a Scalene triangle! 26. Look at the triangle. Find X first by using the Pythagorean Theorem: 52 + ๐ฅ 2 = 132 ๐ฅ = 12 18 13 5 X y Find Y by using the Pythagorean Theorem a second time: 122 + ๐ฆ 2 = 182 ๐ฆ 2 = 180 ๐ฆ = โ180 ๐ฆ = โ(36)(5) ๐ฆ = (โ36)(โ5) ๐ฆ = 6โ5 Determine the length of y. Express your answer in simplified radical form. Write your answer on the line. What is the length of y? ๐โ๐ 38. The figure is a rectangular prism with dimensions 12 inches long, 5 inches wide and 7 inches tall. Determine the length of BI . B C A D 7 G F Write your answer on the line. H 5 I 12 What is the length of BI ?โ218 ๐๐ โ 14.8 We need to use the Pythagorean Theorem TWICE to solve this problem. To begin, draw line segment ฬ ฬ ฬ ๐บ๐ผ to form Right โBGI. Begin by finding the length of ฬ ฬ ฬ ๐บ๐ผ , which is the hypotenuse of Right โGFI (bottom of the rectangular prism) ฬ ฬ ฬ ฬ )2 + (๐น๐ผ ฬ ฬ ฬ )2 ฬ ฬ ฬ )2 = (๐บ๐ผ (๐บ๐น ฬ ฬ ฬ ฬ ฬ ฬ ฬ ๐บ๐น = 5, ๐น๐ผ = 12 Soโฆ.. ฬ ฬ ฬ )2 52 + 122 = (๐บ๐ผ ฬ ฬ ฬ 13 = ๐บ๐ผ Now use the length of ฬ ฬ ฬ ๐บ๐ผ and ฬ ฬ ฬ ฬ ๐ต๐บ to find the length of ฬ ฬ ฬ ๐ต๐ผ using the Pythagorean Theorem a second time on Right โBGI ฬ ฬ ฬ ฬ )2 + (๐บ๐ผ ฬ ฬ ฬ )2 = (๐ต๐ผ ฬ ฬ ฬ )2 (๐ต๐บ ฬ ฬ ฬ ฬ = 7, ฬ ฬ ฬ ๐ต๐บ ๐บ๐ผ = 13 Soโฆ. ฬ ฬ ฬ )2 72 + 132 = (๐ต๐ผ ฬ ฬ ฬ )2 218 = (๐ต๐ผ ๐ต๐ผ โ218 = ฬ ฬ ฬ 14.8 โ ฬ ฬ ฬ ๐ต๐ผ 56. JKLM is an isosceles trapezoid with J(0, โ1), K(โ2, 3) and M(6, โ1). Determine the coordinates of L. 0 E. 0 F. 0 G. 0 H. L (6, 1) L (9, 4) L (2, 3) L (8, -3) Work---see addendum at end of answer key! In order for JKLM to be an isosceles trapezoid, the ฬ ฬ ฬ โ ๐๐ฟ ฬ ฬ ฬ ฬ . To get from point J to length of side ๐ฝ๐พ point K, we go down two and over two. In order to get to point L from point M, we must also travel down two and over (in order to establish the same length) The coordinates of point L will be at (8, โ3) J M K The ratio for a 30°-60°-90° triangle are: 65. Determine the exact value of x in ฮLMN. 0 A. 25 N 60° 2๐ ๐ 25 2 0 C. 25 3 0 B. 30° ๐โ3 50 0 D. 100 x Since the hypotenuse is 50 and equal to 2X, we can set up an equation and solve for X: L 60° M 2๐ฅ = 50 ๐ฅ = 25 The long leg is the value of ๐ฅโ3 So the value of the long leg is 25โ3 66. ฮRST has vertices R(-2 5), S(1, 1), and T(-6, 2). Classify ฮRST based on its sides. 0 A. 0 B. 0 C. 0 D. isosceles scalene right isosceles right Length ฬ ฬ ฬ ฬ ๐ ๐ = โ(โ2 โ 1)2 + (5 โ 1)2 ฬ ฬ ฬ ฬ ๐ ๐ = โ9 + 16 ฬ ฬ ฬ ฬ ๐ ๐ = 5 R T S ฬ ๐๐ ฬ ฬ ฬ = โ(1 โ (โ6))2 + (1 โ 2)2 ฬ ๐๐ ฬ ฬ ฬ = โ49 + 1 ฬ ๐๐ ฬ ฬ ฬ = โ50 ฬ ฬ ฬ ฬ = โ(โ6 โ (โ2))2 + (2 โ 5)2 ๐๐ ฬ ฬ ฬ ฬ ๐๐ = โ16 + 9 ฬ ฬ ฬ ฬ ๐๐ = 5 Since the slopes are opposite reciprocals the lines are perpendicular making it a right triangle. Since two sides are the same length, it is also isosceles. ฬ ฬ ฬ ฬ = ๐๐๐๐๐ ๐ ๐ 5โ1 4 =โ โ2 โ 1 3 ๐๐๐๐๐ ฬ ฬ ฬ ฬ ๐๐ = 2โ5 3 = โ6 โ (โ2) 4