A plane can be names using a capital cursive letter OR using three points, which are not collinear (not on a straight line)


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1 Geometry  Semester 1 Final Review Quadrilaterals (Including some corrections of typos in the original packet) 1. Consider the plane in the diagram. Which are proper names for the plane? Mark all that apply. a. Plane L b. Plane ABC c. Plane DBC d. Plane E e. Plane EDL A plane can be names using a capital cursive letter OR using three points, which are not collinear (not on a straight line) Name: Per: Date: 2. Which diagram does not show plane ABC? a. b. c. d. In c. A, B, C are collinear 3. Which diagram does not show CD bisecting AB? a. b. c. In a. and b. the tickmarks indicate that AB was bisected. Option c. shows AB BISECTING CD, but not BISECTED 4. Which two angles are vertical angles in the diagram? a. AEB and CED a. two straight angles. b. E and E b. ambiguous name for the angles. c. AC and DB c. not a proper way to name any angle. d. two (obtuse) vertical angles. d. AED and CEB 5. Which two angles make a linear pair in the same diagram? a. AEB and CED a. two straight angles this is NOT a linear pair. b. AEC and BED b. vertical angles. c. AED and CEA c. adjacent supplementary angles are a linear pair. d. AED and CEB d. vertical angles.
2 6. In the diagram on the right 1 2 3, and m CEB = 55, m DEL = 65. Which of the following cannot be concluded? a. KLE 1 b. AB FG c. CD KH d. BEL ELK For 6&7: Use vertical angles and linear pairs to figure out measures of angles in the diagram. Remember lines are parallel if and only if alternate interior angles are congruent. (you can use other pairs of angles as well.) 7. In the diagram on the right, lines n and m are cut by transversals p and q. Which value of x would make lines n and m parallel? a. 110 b. 80 c. 70 d o 70o 8. Two angles are supplementary. What can be concluded about them? a. They are both acute. b. They are both obtuse. c. They cannot be both right. d. Either they are both right, or one is acute and one is obtuse. Supplementary = adding up to Two angles are complementary. What can be concluded about them? Complementary = adding to 90 a. They are both acute. b. They are both obtuse. c. They are both right. d. Either they are both right, or one is acute and one is obtuse. 10. If m TUY = 90 and m YUV = 55, find m TUV = 145 (angle addition postulate) 11. If m TUV = 150 and m YUV = 48, find m TUY = 102 (angle addition postulate) 12. If m ABC = 130, m ZBC = x and m ABZ = x + 29, find x. x x + 29 = 130, 2x = 130, 2x = 18, x = Find m ABC if m ABZ = 37x + 1, m ZBC = 110, and m ABC = 147x x = 147x + 1, 110x = 110, x = 1, m ABC = Find JM By segment addition postulate: JM = JK + KM and JK + KL = JL. Therefore, JK=2, JM= A, B, C are collinear and B is between A and C. Find x if AC = 8, BC = x 2, and AB = x 8. x 8 + x 2 = 8, so x=9 16. Find the midpoint between A and B. a. 3 b. 2 c. 1.5 d Which of the following starts with If a transversal crosses two parallel lines? (Mark all that apply) a. Corresponding Angles Postulate f. Converse of Corresponding Angles Postulate b. Alternate Interior Angles Theorem c. Alternate Exterior Angles Theorem g. Converse of Alternate Interior Angles Theorem h. Converse of Alternate Exterior Angles Theorem d. Same Side Interior Angles Theorem i. Converse of Same Side Interior Angles Theorem e. Same Side Exterior Angles Theorem j. Converse of Same Side Exterior Angles Theorem
3 18. Which of the following starts with If the <adjective> angles are? (Mark all that apply) a. Corresponding Angles Postulate f. Converse of Corresponding Angles Postulate b. Alternate Interior Angles Theorem c. Alternate Exterior Angles Theorem g. Converse of Alternate Interior Angles Theorem h. Converse of Alternate Exterior Angles Theorem d. Same Side Interior Angles Theorem i. Converse of Same Side Interior Angles Theorem e. Same Side Exterior Angles Theorem j. Converse of Same Side Exterior Angles Theorem 19. On a separate paper: This should be in your notes, as well as in Google Classroom notes. a. State and prove the alternate interior angles theorem. b. State and prove the converse of the alternate interior angles theorem. * 20. In the right triangle ABC, with m B = 90, the hypotenuse is AC = 16cm. BD AC, and CD = 7cm. Find the length of BD. a. BD = 3 7 b. BD = 4 7 c. BD = 7 3 d. BD = The coordinates of the endpoints of AB are A(0, 0) and B(0, 6). What is the equation of the perpendicular bisector of AB? a. x = 0 b. x = 3 c. y = 0 d. y = The diagram shows isosceles triangle ABC. What is the value of x? a. 10 b. 28 c. 32 d. 40 x+20 x x x = 180, 5x = 140, x = Which of the lines is parallel to the line y = 4x + 7 and passes through the point ( 4,2)? a. y = Z x + 3 c. y = Z x + 2 Slope would be 4 as well. Point slope: y 2 = 4(x + 4), simplify. [ [ Or write y = 4x + b, plug in the point: 2 = b and solve b. y = 4x + 2 d. y = 4x 14 for b. Or check options c.&d. by plugging in the given point. 24. What is the equation of a line passing through (2, 1) and parallel to the line y = 2x + 1? a. y = Z x c. y = 2x 5 Slope would be 2 as well. Point slope: y + 1 = 2(x 2), simplify. \ b. y = Z x + 1 d. y = 2x 1 Or write y = 2x + b, plug in the point: 1 = b and solve for b. Or check options c.&d. by plugging in the given point. \ 25. Which of the following is perpendicular to a line with slope Z ]? a. y = Z ] x b. y = 3x + 5 Using the Pythagorean Theorem: 7 \ + h \ = x \, 9 \ + h \ = y \, and x \ + y \ = 16 \. Subtract the first two equations: 9 \ 7 \ = y \ x \. Add to the third equation: 2y \ = 16 \ + 9 \ 7 \ = 288 So y = 12. Plug in to find h: 9 \ + h \ = 12 \, and get h = 63 = 3 7 c. y = 3x 1 d. y = Z ] x Which is an equation of a perpendicular bisector to PQ, where P( 2,5) and Q(6, 3). a. y = x + 3 c. y = x 1 Slope of PQ is 8/8 = 1, so a perpendicular lines would have slope 1 b. y = x + 3 d. y = x 1 (opposite reciprocal). Midpoint is (2, 1) is on the perpendicular line. Use any of the methods described above to find the equation. 27. Which is an equation of a perpendicular bisector to RS, where R(8,2) and S(0,6). a. y = 2x 4 c. y = Z x + 6 b. y = Z x + 2 \ Slope of RS is 4/8 = 1/2, so a perpendicular lines would have slope 2 \ d. y = 2x 12 (opposite reciprocal). Midpoint is (4, 4) is on the perpendicular line. 28. Point M is the midpoint of AB. If A( 3,6) and M( 5,2), what are the coordinates of B? a. (1,2) b. (7,10) c. (4,4) d. (7,2) x Slope would be 3 (opposite reciprocal) Only one option fits this. M is 2 units left and 4 units down from A. Therefore, B is 2 units left and 4 units down from M, at ( 7, 2) h 9 cm y 21. AB is vertical, so the perpendicular bisector is horizontal, and passes through the midpoint (0,3)
4 29. In the diagram on the right A g B g C g is the image of ABC and A gg B gg C gg is the image of A B C. The combined transformation mapping ABC onto A gg B gg C gg is an example of: a. Reflection followed by a rotation. b. Reflection followed by a translation. c. Translation followed by a rotation. d. Translation followed by a reflection. 30. Describe the transformation mapping ABC onto A B C specifically (verbally, or using function notation). Translate 2 units to the right and 1 unit up, then reflect across x=1. Alternatively: r klz (T (\,Z) ). Or: (x, y) (x + 2, y + 1) followed by (x, y) (2 x, y) 31. Parallelogram ABCD has vertices A 2,0, B 0, 3, C 3, 3, and D(5,0) a. Use slopes to show that it is indeed a parallelogram. Slope of AB and of CD: 3/2, slope of BC and of DA: 0. Opposite sides have equal slopes, so they are parallel. By definition ABCD is a parallelogram. b. If it is reflected across the xaxis, how many vertices would stay at their original position? Vertices that would not be affected are on the line of reflection, so ycoordinate is 0: two vertices. 32. What are the coordinates of (3,4) reflected across the line y = x? a. ( 4, 3) b. ( 3,4) c. (4, 3) d. (4,3) 33. In FGH, m F = 41 and m H = 102. What is m G? 180 ( ) = 37 a. 47 b. 61 c. 143 d ACD is isosceles with AC CD. If m A = 36 and m BCD = 19. Find m B. a. 17 b. 19 c. 36 d In DEF, m D = 3x + 5, m E = 4x 15 and m F = 2x Which statement is true? a. DF FE b. E F c. DE FE d. D F x x x + 10 = 180, so 9x = 180, and x = 20 Therefore, m D = 65, m E = 65. The angles are congruent, meaning DF FE 36. In the diagram on the right, which transformation maps GH onto G H? a. Reflection across the yaxis. Many possible answers, but not of b. Rotation 180 about the origin. the answers listed: c. Reflection across the xaxis. Translate 6 to the right and then d. Translation x, y (x + 6, y 8) reflect across the xaxis. 37. When writing a proof, which of the following relationships between angles can imply that the angles are congruent? a. Supplementary angles b. Linear pair c. Adjacent angles d. Vertical angles
5 38. In the diagram, AC bisects BAD, and B D. Which criterion can be used to prove ABC ADC? a. SSS c. SAS b. AAA d. AAS 39. Can a triangle have two right angles? Explain. 40. Which of the following can be used to prove that a parallelogram is a rhombus? a. The diagonals are congruent. b. The opposite sides are parallel. c. The diagonals are perpendicular. d. The opposite angles are congruent. 41. Given parallelogram BIRD, find the values of x and y. a. x = 2, y = 10 b. x = 10, y = 2 17x 59 = 8x + 31, so 9x = 90, x = 10 c. x = 31, y = 111 d. x = 111, y = 31 12y + 7 = 48y 65, so 36y = 72, y = Which statement is not sufficient to prove that ABCD is a parallelogram? a. AD DC and AB BC option a. would mean ABCD is a kite. b. AC and BD bisect each other c. A C and B D d. AB DC and AB DC 43. Given quadrilateral BIKE, MI = 7x + 2, ME = 6x + 9. What value of x makes BIKE a parallelogram? a. 7 c. 51 7x + 2 = 6x + 9, so x = b d Which statement is false? a. A quadrilateral is a square if and only if it is a rhombus and a rectangle. b. A quadrilateral is a rectangle if and only if it has four right angles. c. A parallelogram is a rectangle if and only if its diagonals are congruent. d. A parallelogram is a rhombus if and only if its diagonals are congruent. 45. If quadrilateral WIND is a rectangle, find the length of WN a. 8 b. 16 The diagonals of a rectangle are congruent, c. 24 d. 32 and also bisect each other. 46. What is the most specific name for the quadrilateral CATS? a. Rectangle b. Parallelogram c. Trapezoid d. Isosceles trapezoid No. Two right angles add to 180, so the third angle would be 0, which means the shape is not a triangle. This can be either a parallelogram with a right angle or an isosceles 47. Which statement about kites is false? trapezoid. However, in the latter case, the base angles would be a. A kite s diagonals are perpendicular. congruent, so all angles are 90, giving a rectangle again. b. A kite s opposite sides are congruent. c. A kite has two pairs of consecutive congruent sides. d. A kite has exactly one pair of opposite angles that are congruent.
6 48. Find the length of the midsegment of the trapezoid shown. a. 19 c b. 20 d. 22 = The diagram of the right shows kite SURF. Find the measure of S. a. 95 c. 163 b d Which statement about an isosceles trapezoid is false? a. The legs are congruent. b. It has only one pair of congruent base angles. c. Its diagonals are congruent. d. Both pairs of base angles are congruent. 51. Find the value of the variables in the tall parallelogram on the right. a. x = 2, y = 2 3x + 2 = 5x 2 c. x = 1, y = 3.2 b. x = 5, y = 2 9x + 2 = 5y 5 d. x = 2, y = Find the value of the variables in the parallelogram on the right. a. x = 4, y = 184 c. x = 3, y = 184 3x + 4 = 16 b. x = 4, y = 116 y 60 = 56 d. x = 3, y = Find the value of the variables in the parallelogram. a. n = 4, m = 5 b. n = 6, m = 3 2n 1 = 9 3m = m + 8 c. n = 5, m = 4 d. n = 2, m = In ABC, AD is a perpendicular bisector of BC. Draw a diagram showing this information and prove that ABC is isosceles. (see below.) 55. ABCD is a parallelogram with A 2,3 B 5,7 C 3,6. Find point D. a. D(0, 10) b. D( 10, 4) c. D(0.5, 4.5) d. D 6,2 56. The diagonals of rhombus PQRS below intersect at T. Given that m RPS = 30 and RT = 6, find a. m QPR a. 30, b/c in a rhombus diagonals bisect angles. b. m QTP b. 90, b/c in a rhombus diagonals are perpendicular c. 12, because the diagonals bisect each other. c. RP Statements AD is a perpendicular bisector of BC CD BD ADC, ADB are right angles ADC ADB AD AD ADC ADB AC AB Reasons Given Definition of bisector Definition of perpendicular lines All right angles are congruent Reflexive property SAS CPCTC 57. The diagonals of rectangle WXYZ on the right intersect at P. Given that m YXZ = 50 and XZ = 12, find a. m WXZ b. m WPX c. PY a = 40, b/c WXY = 90 b = 100, b/c the diagonals are congruent and bisecting each other, so WPX is isosceles c. 6, because the diagonals bisect each other.
7 58. Find the value of x. a. b. c. a. 4x = 8, so x = 2 b. 2x 1 = [[qzr = 27, 2x = 28, x = 14 \ c. 3x 2 = (2x + 2)/2, 3x 2 = x + 1, x = 1.5 * 59. How many angles measuring 58 are there in the diagram on the right? a. 1 b. 2 c. 5 d. 6 Corresponding angles to D with parallel lines: EAB, BCF Alternate int. angle with BCF, gives also ABC. Using the isosceles triangle in the diagram, we get E. Finally, using sum of angles in a triangle and a linear pair we get CBF is also Use the information in the diagram to find m A, m D, m DNA Since DNA is an isosceles triangle, m D = Zsrt\k = 90 x Since we have parallel line: 2x + 3x x = 180 4x = 80, x = 20, m A = m D = 70, m DNA = 40 \ 58 o 58 o 58 o 58 o 58 o 61. Nanorta puts up a tent with a crosssection of an isosceles triangle on top of an isosceles trapezoid, as illustrated in the diagram. She measures m A = 62 and m E = 140. Help her figure out m BDE =118 o 20 o 20 o 62. In the previous question, which of the following properties or theorems is NOT used? a. Vertical angles theorem. c. Angle addition postulate. b. Triangle sum theorem. d. Same side interior angles theorem. 63. Which set of statements is NOT enough to show that AED BEC? a. D C, AE BE, AED BEC b. A B, AE BE, AED BEC c. AE BE, AED BEC, DE CE d. AE BE, AED BEC, AD BC Option d. is SSA 64. Use the information in the diagram to prove that ABC is isosceles. Statements Reasons Follow these steps: AD AE, BD EC Given a. What is the given information? DAE is isosceles b. Explain why Isosc. theorem c. Explain why Supp. to s d. Show ABD ACE. ADB AEC SAS e. Conclude AB AC. AB AC CPCTC ABC is isosceles Definition of isosc. Definition of isosc. 65. Find KM. 66. Find FE. 2*10=20 2x 12 = 2x 8 = x 4 2 x = 8
8 67. ABCD is a square. E and F are points on BC and DC respectively. Given that BE DF, complete the proof that ABE ADF. D B SAS DF BE 68. For each pair of triangles, determine if they can be proven congruent. If so, identify the congruence criterion that proves they are congruent and write the congruence statement. If not, write inconclusive. a. b. c. ABC AKC by AAS DAB BCD by SSS DAB DCB by SSS d. e. f. KLP MNP by ASA inconclusive inconclusive (SSA) 69. Consider the two triangles in the diagram on the right. a. Can you use the vertical angles theorem to show that AWT PWS? Explain. No. They are NOT vertical angles. b. How can you show that WAT WPS? Supplementary to congruent angles c. Which criterion would you use to prove WAT WPS? ASA d. What kind of triangles are WAT and WPS? What theorem are you relying on? e. Find m AWP. m AWT = m PWS = 20, m AWP = = 140 Isosceles, by the converse of the isosceles triangle theorem. 70. Show that ABCD with A 0,3, B 3,7, C 8,7, D(11,3) is an isosceles trapezoid. Slope of AD and BC is 0, so the sides are parallel. Slopes of AB = 4/3 and CD = 4/3, so not parallel, hence a trapezoid. Leg lengths are equal: 3 \ + 4 \ =5 71. Parallelogram ABCD has coordinates A 1,5, B 6,3, C(3, 1). a. Find the coordinates of D. From A to B: 5 right and 2 down. So, from C to D, 5 left and 2 up: ( 2,1) b. What are the coordinates of the point of intersection of the diagonals AC and BD? Midpoint of AC: (2,2)
9 72. Complete the proof that opposite angles in a parallelogram are congruent. Given: ABCD Prove: B D Statements ABCD AB CD, BAC DCA DAC BCA AC AC BC AD ABC CDA B D Reasons Given Definition of parallelogram Alternate interior angle theorem Alternate interior angle theorem Reflexive property ASA CPCTC 73. For each property, specify all quadrilateral that satisfy it. (Write T for trapezoid, IT for isosceles trapezoid, K for kite, P for parallelogram, R for rectangle, D for rhombus/diamond, and S for square) a. The diagonals are congruent. a. IT, R b. Both pairs of opposite sides are congruent. b. P, R, D, S c. Both pairs of opposite sides are parallel. c. P, R, D, S d. All angles are congruent. d. R, S e. All sides are congruent. e. D, S f. Diagonals bisect each other. f. P, R, D, S g. Diagonals bisect the angles. g. D (for kites, only one diagonal does) h. Diagonals are perpendicular to each other. h. D, K i. One pair of congruent opposite sides. i. K 74. Which statement is incorrect? a. In an isosceles trapezoid, the diagonals are congruent. b. In an isosceles trapezoid, opposite angles are supplementary. c. In an isosceles trapezoid, a diagonal forms two congruent triangles. d. In an isosceles trapezoid, the diagonals form four small triangles, two of which are congruent. 75. Parallelograms MNOP and OPQR share a side. Prove that MN QR Opposite sides are congruent Given OP QR Opposite sides are congruent Transitive property
10 76. Use slopes of sides and diagonals to determine the most specific type of quadrilateral ABCD. a. A 0,3, B 3,0, C 6, 9, D( 9, 6) Slope of AB and CD is 1 and slope of the other two sides is 1. Opposite reciprocal slopes, means it is a rectangle. Slopes of diagonals are: AC = 2 and BD=1/2. NOT opposite reciprocal, so diagonals are not perpendicular, and it is not a square. RECTANGLE b. A 0,8, B 3,4, C 3,9, D(0,13) Opposite sides have equal slopes: undefined and 4/3. Not opposite reciprocal, so not a rectangle or square. Slopes of diagonals are 1/3 and 3, so diagonals are perpendicular. Therefore, this is a RHOMBUS c. A 2,5, B 3, 2, C 8,3, D 6,7 Opposite sides do not have same slope(1, ½, 2, 7), so not a parallelogram nor a trapezoid. Slopes of diagonals are 3 and 1/3, so the diagonals are perpendicular. Hence, this is a KITE d. A 1,1, B 4,5, C 9, 7, D(6, 11) Opposite slopes are 4/3 and 4/3 (parallel sides) and 12/5 and 12/5 (parallel sides), so a parallelogram but not a rectangle. Slopes of diagonals are 1 and 8, so the diagonals are not perpendicular. Therefore, a PARALLELOGRAM 77. Complete the flowchart proof. Given: RHOM is a rhombus, OB RM, RU MO. Prove: OB RU OBM RUM Definition of perpendicular lines All sides of a rhombus are congruent OBM RUM All right angles are congruent OBM AAS RUM Reflexive property CPCTC 78. Quadrilateral ABCD is a rhombus. a. If m BAE = 32, find m ECD. b. If m EDC = 43, find m CBA c. If m EAB = 57, find m ADC a. 32 (alt. int. angles) b. 86 (opp. s are and bisected by the diagonal) c = 66 (diagonal bisect and consecutive angles are supplementary. d. Find x given that m BEC = 3x 15. e. Find x given that m ADE = 5x 8 and m CBE = 3x + 24 d. 3x 15 = 90, 3x = 105, x = 35 e. 5x 8 = 3x + 24, 2x = 32, x = 16 (alt. int. angles)
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