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Final Review Geometry Multiple Choice Identify the choice that best completes the statement or answers the question. ____ 1. Which is a correct two-column proof? Given: Prove: and n are supplementary. p d l b c h j a. b. c. k m d. none of these ____ 2. . Find the value of x for p to be parallel to q. The diagram is not to scale. 3 4 5 1 2 6 p q a. 114 ____ b. 126 c. 120 d. 20 c. 147 d. 75 3. Find the value of x. The diagram is not to scale. 72° 105° a. 33 ____ x° b. 162 4. The folding chair has different settings that change the angles formed by its parts. Suppose is 70. Find . The diagram is not to scale. 1 2 3 a. 96 ____ b. 106 c. 116 5. Find the value of x. The diagram is not to scale. Given: , , d. 86 is 26 and S R a. 5 ____ T U b. 24 c. 20 d. 40 c. pentagon d. octagon 6. Classify the polygon by its sides. a. triangle b. hexagon ____ 7. Complete this statement. The sum of the measures of the exterior angles of an n-gon, one at each vertex, is ____. a. (n – 2)180 b. 360 c. d. 180n ____ 8. Write an equation for the horizontal line that contains point E(–3, –1). a. x = –1 b. x = –3 c. y = –1 d. y = –3 ____ 9. Which two lines are parallel? I. II. III. a. I and II b. I and III c. II and III d. No, two of the lines are parallel. ____ 10. What is the converse of the following conditional? If a point is in the first quadrant, then its coordinates are positive. a. If a point is in the first quadrant, then its coordinates are positive. b. If a point is not in the first quadrant, then the coordinates of the point are not positive. c. If the coordinates of a point are positive, then the point is in the first quadrant. d. If the coordinates of a point are not positive, then the point is not in the first quadrant. ____ 11. Which conditional has the same truth value as its converse? a. If x = 7, then . b. If a figure is a square, then it has four sides. c. If x – 17 = 4, then x = 21. d. If an angle has measure 80, then it is acute. ____ 12. If possible, use the Law of Detachment to draw a conclusion from the two given statements. If not possible, write not possible. Statement 1: If x = 3, then 3x – 4 = 5. Statement 2: x = 3 a. 3x – 4 = 5 c. If 3x – 4 = 5, then x = 3. b. x = 3 d. not possible ____ 13. In the paper airplane, and D H A B Find E C G Drawing not to scale a. 131 F b. 49 c. 90 d. 59 ____ 14. The two triangles are congruent as suggested by their appearance. Find the value of c. The diagrams are not to scale. d° 38° g 5 b f° e° 3 a. 4 4 52° c b. 5 ____ 15. Name the angle included by the sides c. 3 and d. 38 N M P a. b. c. d. none of these ____ 16. Name the theorem or postulate that lets you immediately conclude | A D | B C a. SAS b. ASA c. AAS d. none of these ____ 17. What is the measure of the vertex angle of an isosceles triangle if one of its base angles measures 42°? a. 69° b. 84° c. 138° d. 96° ____ 18. Which overlapping triangles are congruent by ASA? a. b. c. d. ____ 19. Name the point of concurrency of the angle bisectors. a. A b. B c. C d. not shown ____ 20. Which diagram shows a point P an equal distance from points A, B, and C? a. c. b. d. ____ 21. For the triangle, find the coordinates of the point of concurrency of the perpendicular bisectors of the sides. y 5 –5 5 x –5 a. (1, 1) b. 3 3 ( , ) 2 2 c. 1 ( , 1) 2 d. 3 1 ( , ) 2 2 bisects DAB . Find ED if ____ 22. Given: a. 51 b. 540 and (not drawn to scale) c. 39 d. 21 ____ 23. Which of these lengths could be the sides of a triangle? a. 15 cm, 4 cm, 20 cm c. 11 cm, 5 cm, 16 cm b. 3 cm, 15 cm, 20 cm d. 5 cm, 12 cm, 16 cm ____ 24. Which statement is true? a. All quadrilaterals are rectangles. b. All quadrilaterals are squares. c. All rectangles are quadrilaterals. d. All quadrilaterals are parallelograms. ____ 25. In parallelogram DEFG, DH = x + 3, HF = 3y, GH = 4x – 5, and HE = 2y + 3. Find the values of x and y. The diagram is not to scale. D E H G a. x = 6, y = 3 F b. x = 2, y = 3 c. x = 3, y = 2 ____ 26. What is the missing reason in the proof? Given: parallelogram ABCD with diagonal Prove: d. x = 3, y = 6 A B D C Statements Reasons 1. Definition of parallelogram 1. 2. 3. 4. 5. 6. 2. Alternate Interior Angles Theorem 3. Definition of parallelogram 4. Alternate Interior Angles Theorem 5. Reflexive Property of Congruence 6. ? a. Reflexive Property of Congruence b. ASA c. Alternate Interior Angles Theorem d. SSS ____ 27. DEFG is a rectangle. DF = 5x – 5 and EG = x + 11. Find the value of x and the length of each diagonal. a. x = 4, DF = 13, EG = 13 c. x = 4, DF = 15, EG = 15 b. x = 4, DF = 15, EG = 18 d. x = 2, DF = 13, EG = 13 ____ 28. Find in the kite. The diagram is not to scale. | 3 1 2 B || || D 39° | A C a. 51, 51 b. 39, 39 c. 39, 51 ____ 29. Name the set(s) of numbers to which –5 belongs. a. whole numbers, natural numbers, integers b. rational numbers c. whole numbers, integers, rational numbers d. integers, rational numbers Complete the statement with , , or =. ____ 30. d. 51, 39 a. b. c. = Short Answer 31. Give the missing reasons in this proof of the Alternate Interior Angles Theorem. Given: Prove: 32. The 8 rowers in the racing boat stroke so that the angles formed by their oars with the side of the boat all stay equal. Explain why their oars on either side of the boat remain parallel. 33. Identify the form of the equation –3x – y = –2. To graph the equation, would you use the given form or change to another form? Explain. 34. Explain how you can use SSS, SAS, ASA, or AAS with CPCTC to complete a proof. Given: Prove: B C A D 35. In the figure, , , and . Prove that . 36. Can these three segments form the sides of a triangle? Explain. c b a In the diagram, are midsegments of triangle ABC. Find the value of the variable if . 37. z 38. What type of quadrilateral has exactly one pair of parallel sides? 39. The fact that the diagonals of a kite are perpendicular suggests a way to place a kite in the coordinate plane. Show this placement. Include labels for the kite vertices. Approximate the square root to the nearest whole number. 40. Essay 41. Write a paragraph proof of this theorem: In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. Given: Prove: s 1 2 3 4 5 6 7 8 r t 42. Write a two-column proof. Given: Prove: are supplementary. 1 2 3 l 4 5 6 7 8 m 43. Find the values of the variables. Show your work and explain your steps. The diagram is not to scale. o 31 x w v y o 68 z 44. Given: are complementary, and are complementary. Prove: Other 45. Is each figure a polygon? If yes, describe it as concave or convex and classify it by its sides. If not, tell why. a. b. c. 46. Line p contains points A(–1, 4) and B(3, –5). Line q is parallel to line p. Line r is perpendicular to line q. What is the slope of line r? Explain. 47. When you open a stepladder, you use a brace on each side of the ladder to lock the legs in place. Explain why the triangles formed on each side by the legs and the ground ( in the diagram) are congruent. 48. A tennis court has a baseline at each end. One is labeled in the picture. Which part of the tennis court is equidistant from the midpoints of the two baselines? Explain. Final Review Geometry Answer Section MULTIPLE CHOICE 1. ANS: OBJ: TOP: KEY: 2. ANS: REF: OBJ: NAT: KEY: 3. ANS: REF: OBJ: NAT: TOP: 4. ANS: REF: OBJ: NAT: TOP: 5. ANS: REF: OBJ: NAT: KEY: 6. ANS: REF: NAT: TOP: 7. ANS: REF: NAT: KEY: 8. ANS: OBJ: NAT: TOP: 9. ANS: REF: NAT: TOP: 10. ANS: OBJ: TOP: 11. ANS: A PTS: 1 DIF: L2 REF: 3-1 Properties of Parallel Lines 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1 3-1 Example 3 proof | two-column proof | supplementary angles | parallel lines | reasoning D PTS: 1 DIF: L2 3-3 Parallel and Perpendicular Lines 3-3.1 Relating Parallel and Perpendicular Lines NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1 TOP: 3-3 Example 2 parallel lines A PTS: 1 DIF: L2 3-4 Parallel Lines and the Triangle Angle-Sum Theorem 3-4.2 Using Exterior Angles of Triangles NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 3-4 Example 3 KEY: triangle | sum of angles of a triangle A PTS: 1 DIF: L2 3-4 Parallel Lines and the Triangle Angle-Sum Theorem 3-4.2 Using Exterior Angles of Triangles NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 3-4 Example 4 KEY: triangle | sum of angles of a triangle D PTS: 1 DIF: L3 3-4 Parallel Lines and the Triangle Angle-Sum Theorem 3-4.2 Using Exterior Angles of Triangles NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 exterior angle B PTS: 1 DIF: L2 3-5 The Polygon Angle-Sum Theorems OBJ: 3-5.1 Classifying Polygons NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 3-5 Example 2 KEY: classifying polygons B PTS: 1 DIF: L2 3-5 The Polygon Angle-Sum Theorems OBJ: 3-5.2 Polygon Angle Sums NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 Polygon Exterior Angle-Sum Theorem C PTS: 1 DIF: L2 REF: 3-6 Lines in the Coordinate Plane 3-6.2 Writing Equations of Lines NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 3-6 Example 6 KEY: vertical line A PTS: 1 DIF: L2 3-7 Slopes of Parallel and Perpendicular Lines OBJ: 3-7.1 Slope and Parallel Lines NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 3-7 Example 2 KEY: slopes of parallel lines | parallel lines C PTS: 1 DIF: L2 REF: 2-1 Conditional Statements 2-1.2 Converses NAT: NAEP 2005 G5a 2-1 Example 5 KEY: conditional statement | coverse of a conditional C PTS: 1 DIF: L2 REF: 2-1 Conditional Statements 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. OBJ: TOP: KEY: ANS: OBJ: KEY: ANS: OBJ: TOP: ANS: OBJ: TOP: ANS: REF: OBJ: TOP: ANS: REF: OBJ: TOP: ANS: REF: OBJ: NAT: KEY: ANS: REF: OBJ: TOP: KEY: ANS: REF: NAT: KEY: ANS: REF: NAT: KEY: ANS: LOC: TOP: KEY: NOT: ANS: NAT: KEY: ANS: TOP: MSC: ANS: 2-1.2 Converses NAT: NAEP 2005 G5a 2-1 Example 6 conditional statement | coverse of a conditional | truth value A PTS: 1 DIF: L4 REF: 2-3 Deductive Reasoning 2-3.1 Using the Law of Detachment NAT: NAEP 2005 G5a Law of Detachment | deductive reasoning B PTS: 1 DIF: L2 REF: 4-1 Congruent Figures 4-1.1 Congruent Figures NAT: NAEP 2005 G2e | ADP K.3 4-1 Example 2 KEY: congruent figures | corresponding parts C PTS: 1 DIF: L2 REF: 4-1 Congruent Figures 4-1.1 Congruent Figures NAT: NAEP 2005 G2e | ADP K.3 4-1 Example 1 KEY: congruent figures | corresponding parts A PTS: 1 DIF: L2 4-2 Triangle Congruence by SSS and SAS 4-2.1 Using the SSS and SAS Postulates NAT: NAEP 2005 G2e | ADP K.3 4-2 Example 2 KEY: angle A PTS: 1 DIF: L2 4-3 Triangle Congruence by ASA and AAS 4-3.1 Using the ASA Postulate and the AAS Theorem NAT: NAEP 2005 G2e | ADP K.3 4-3 Example 3 KEY: ASA | AAS | SAS D PTS: 1 DIF: L2 4-5 Isosceles and Equilateral Triangles 4-5.1 The Isosceles Triangle Theorems NAEP 2005 G3f | ADP J.5.1 | ADP K.3 TOP: 4-5 Example 2 isosceles triangle | Isosceles Triangle Theorem | Triangle Angle-Sum Theorem | word problem A PTS: 1 DIF: L3 4-7 Using Corresponding Parts of Congruent Triangles 4-7.1 Using Overlapping Triangles in Proofs NAT: NAEP 2005 G3f | ADP K.3 4-7 Example 2 congruent figures | corresponding parts | overlapping triangles | proof C PTS: 1 DIF: L2 5-3 Concurrent Lines, Medians, and Altitudes OBJ: 5-3.2 Medians and Altitudes NAEP 2005 G3b angle bisector | incenter of the triangle | point of concurrency A PTS: 1 DIF: L2 5-3 Concurrent Lines, Medians, and Altitudes OBJ: 5-3.1 Properties of Bisectors NAEP 2005 G3b TOP: 5-3 Example 2 circumcenter of the triangle | circumscribe A PTS: 1 DIF: Level B REF: MHGM0087 NCTM.PSSM.00.MTH.9-12.GEO.2.a Lesson 5.2 Use Perpendicular Bisectors triangle | perpendicular bisector | concurrency MSC: Knowledge 978-0-547-31534-8 C PTS: 1 DIF: Level B REF: PHGM0420 NT.CCSS.MTH.10.9-12.G-SRT.8 TOP: Lesson 5.3 Use Angle Bisectors of Triangles solve | angle bisector MSC: Application NOT: 978-0-547-31534-8 D PTS: 1 DIF: Level B REF: PHGM0418 Lesson 5.5 Use Inequalities in a Triangle KEY: triangle inequality Comprehension NOT: 978-0-547-31534-8 C PTS: 1 DIF: L2 REF: 6-1 Classifying Quadrilaterals 25. 26. 27. 28. 29. 30. OBJ: KEY: ANS: OBJ: TOP: KEY: ANS: OBJ: KEY: ANS: OBJ: TOP: ANS: OBJ: TOP: ANS: OBJ: NAT: TOP: ANS: TOP: KEY: 6-1.1 Classifying Special Quadrilaterals NAT: NAEP 2005 G3f reasoning | kite | parallelogram | quadrilateral | rectangle | rhombus | special quadrilaterals C PTS: 1 DIF: L2 REF: 6-2 Properties of Parallelograms 6-2.2 Properties: Diagonals and Transversals NAT: NAEP 2005 G3f 6-2 Example 3 transversal | diagonal | parallelogram | Theorem 6-3 | algebra B PTS: 1 DIF: L3 REF: 6-2 Properties of Parallelograms 6-2.2 Properties: Diagonals and Transversals NAT: NAEP 2005 G3f proof | two-column proof | parallelogram | diagonal C PTS: 1 DIF: L2 REF: 6-4 Special Parallelograms 6-4.1 Diagonals of Rhombuses and Rectangles NAT: NAEP 2005 G3f 6-4 Example 2 KEY: rectangle | algebra | Theorem 6-11 | diagonal C PTS: 1 DIF: L2 REF: 6-5 Trapezoids and Kites 6-5.1 Properties of Trapezoids and Kites NAT: NAEP 2005 G3f 6-5 Example 3 KEY: kite | Theorem 6-17 | diagonal D PTS: 1 DIF: L2 REF: 1-3 Exploring Real Numbers 1-3.1 Classifying Numbers NAEP 2005 N1d | NAEP 2005 N1g | NAEP 2005 N1j | ADP I.2.1 | ADP I.2.2 | ADP I.3 1-3 Example 1 KEY: integers | rational numbers B PTS: 1 DIF: Level B REF: MLC30521 Lesson 9.2 Rational and Irrational Numbers irrational numbers | rational numbers | compare | order NOT: 978-0-618-73965-3 SHORT ANSWER 31. ANS: a. Corresponding angles. b. Vertical angles. c. Transitive Property. PTS: 1 DIF: L2 REF: 3-1 Properties of Parallel Lines OBJ: 3-1.2 Properties of Parallel Lines NAT: NAEP 2005 M1f | ADP K.2.1 TOP: 3-1 Example 3 KEY: alternate interior angles | Alternate Interior Angles Theorem | proof | reasoning | two-column proof | multi-part question 32. ANS: The rowers keep corresponding angles congruent. PTS: 1 DIF: L3 REF: 3-2 Proving Lines Parallel OBJ: 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3 TOP: 3-2 Example 1 KEY: transversal | word problem | reasoning | parallel lines 33. ANS: Standard form. Answer may vary. Sample: You could use the given form. Find the intercepts and use them to draw the line. PTS: OBJ: NAT: KEY: 1 DIF: L3 REF: 3-6 Lines in the Coordinate Plane 3-6.2 Writing Equations of Lines NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 graphing | point-slope form | standard form of a linear equation | slope-intercept form | writing in math 34. ANS: Answers may vary. Sample: Since the two triangles share the side by CPCTC. , they are congruent by SAS. Then PTS: 1 DIF: L2 REF: 4-4 Using Congruent Triangles: CPCTC OBJ: 4-4.1 Proving Parts of Triangles Congruent NAT: NAEP 2005 G2e | ADP K.3 TOP: 4-4 Example 2 KEY: SAS | CPCTC | writing in math | reasoning 35. ANS: Answers may vary. Sample: by SAS, so . Supplements of congruent angles are congruent, so . by AAS. PTS: 1 DIF: L4 REF: 4-7 Using Corresponding Parts of Congruent Triangles OBJ: 4-7.1 Using Overlapping Triangles in Proofs NAT: NAEP 2005 G3f | ADP K.3 KEY: overlapping triangles | proof | AAS 36. ANS: No; for three segments to form the sides of a triangle, the sum of the length of two segments must be greater than the length of the third segment. PTS: 1 DIF: L3 REF: 5-5 Inequalities in Triangles OBJ: 5-5.2 Inequalities Involving Sides of Triangles NAT: NAEP 2005 G3f KEY: Triangle Inequality Theorem 37. ANS: 15 PTS: 1 DIF: Level B REF: 7f580833-cdbb-11db-b502-0011258082f7 TOP: Lesson 5.1 Midsegment Theorem and Coordinate Proof KEY: Midsegment theorem MSC: Knowledge NOT: 978-0-547-31534-8 38. ANS: trapezoid PTS: 1 DIF: L2 REF: 6-1 Classifying Quadrilaterals OBJ: 6-1.1 Classifying Special Quadrilaterals NAT: NAEP 2005 G3f KEY: quadrilateral | reasoning | algebra | trapezoid | rhombus | square | parallelogram 39. ANS: Answers may vary. Sample: y (0, c) (–a, 0) (b, 0) x (0, –c) PTS: 1 DIF: L3 OBJ: 6-6.1 Naming Coordinates KEY: kite | algebra | coordinate plane REF: 6-6 Placing Figures in the Coordinate Plane NAT: NAEP 2005 G4d 40. ANS: 11 PTS: 1 DIF: Level B TOP: Lesson 9.1 Square Roots MSC: Comprehension REF: MLC30508 NAT: NT.CCSS.MTH.10.8.8.EE.2 KEY: square roots | approximate NOT: 978-0-618-73965-3 ESSAY 41. ANS: [4] [3] [2] [1] PTS: OBJ: NAT: KEY: 42. ANS: [4] [3] [2] [1] PTS: OBJ: KEY: angles 43. ANS: [4] [3] [2] [1] By the definition of , r s implies m2 = 90, and t s implies m6 = 90. Line s is a transversal. 2 and 6 are corresponding angles. By the Converse of the Corresponding Angles Postulate, r || t. correct idea, some details inaccurate correct idea, not well organized correct idea, one or more significant steps omitted 1 DIF: L4 REF: 3-3 Parallel and Perpendicular Lines 3-3.1 Relating Parallel and Perpendicular Lines NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.1 TOP: 3-3 Example 2 paragraph proof | proof | reasoning | extended response | rubric-based question | perpendicular lines correct idea, some details inaccurate correct idea, some statements missing correct idea, several steps omitted 1 DIF: L4 REF: 3-2 Proving Lines Parallel 3-2.1 Using a Transversal NAT: NAEP 2005 M1e | NAEP 2005 M1f | ADP K.2.3 two-column proof | proof | extended response | rubric-based question | parallel lines | supplementary w + 31 + 90 = 180, so w = 59º. Since vertical angles are congruent, y = 59º. Since supplementary angles have measures with sum 180, x = v = 121º. z + 68 + y = z + 68 + 59 = 180, so z = 53º. small error leading to one incorrect answer three correct answers, work shown two correct answers, work shown PTS: 1 DIF: L3 REF: 3-4 Parallel Lines and the Triangle Angle-Sum Theorem OBJ: 3-4.2 Using Exterior Angles of Triangles NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 KEY: Triangle Angle-Sum Theorem | vertical angles | supplementary angles | extended response | rubric-based question 44. ANS: [4] By the definition of complementary angles, and . By the Transitive Property of Equality (or Substitution Property), . By the Subtraction Property of Equality, , and by the definition of congruent angles. OR equivalent explanation [3] one step missing OR one incorrect justification [2] two steps missing OR two incorrect justifications [1] correct steps with no explanations PTS: 1 DIF: L4 REF: 2-5 Proving Angles Congruent OBJ: 2-5.1 Theorems About Angles NAT: NAEP 2005 G3g | ADP K.1.1 KEY: complementary angles | Properties of Equality | rubric-based question | extended response | proof OTHER 45. ANS: a. concave hexagon b. concave dodecagon c. not a polygon; two sides intersect between endpoints PTS: 1 DIF: L3 REF: 3-5 The Polygon Angle-Sum Theorems OBJ: 3-5.1 Classifying Polygons NAT: NAEP 2005 G3b | NAEP 2005 G3f | ADP J.5.1 | ADP K.1.2 KEY: classifying polygons | convex | concave | writing in math 46. ANS: 4 ; Line r is perpendicular to line p because a line perpendicular to one of two parallel lines is also 9 perpendicular to the other. Thus, the slope of line r is the opposite reciprocal of the slope of line p. PTS: 1 DIF: L3 REF: 3-7 Slopes of Parallel and Perpendicular Lines OBJ: 3-7.2 Slope and Perpendicular Lines NAT: NAEP 2005 A1h | NAEP 2005 A2a | ADP J.4.1 | ADP J.4.2 | ADP K.10.2 KEY: perpendicular lines | parallel lines | slopes of parallel lines | slopes of perpendicular lines | reasoning | writing in math 47. ANS: Answers may vary. Sample: Each leg on one side of the ladder is the same length as the corresponding leg on the other side. The locking braces hold the legs apart by the same angle measure. The triangles are congruent by SAS. PTS: 1 DIF: L3 REF: 4-2 Triangle Congruence by SSS and SAS OBJ: 4-2.1 Using the SSS and SAS Postulates NAT: NAEP 2005 G2e | ADP K.3 KEY: SAS | word problem | problem solving | writing in math 48. ANS: The net; the net is the perpendicular bisector of the segment joining the midpoints of the two baselines. By the Perpendicular Bisector Theorem, any point on the net is equidistant from the midpoints of the two baselines. PTS: OBJ: TOP: KEY: 1 DIF: L2 REF: 5-2 Bisectors in Triangles 5-2.1 Perpendicular Bisectors and Angle Bisectors NAT: NAEP 2005 G3b | ADP K.2.2 5-2 Example 1 Perpendicular Bisector Theorem | perpendicular bisector | reasoning | writing in math