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Fall Geometry Final Review Short Answer 1. When a conditional and its converse are true, you can combine them as a true ____. 2. Write the two conditional statements that make up the following biconditional. I drink juice if (and only if) it is breakfast time. 3. Is the following definition of dog reversible? If yes, write it as a true biconditional. A dog is a mammal. 4. One way to show that a statement is NOT a good definition is to find a ____. 5. Are C, B, and A collinear? If so, name the line on which they lie. C B D A 6. What is the name of the ray that is opposite H G F E 7. Complete the two-column proof. Given: Prove: ? 8. Let p be “you are a senior” and let q be “you are in high school.” Write the converse. Then decide whether it is true or false. 9. If , find the value of FG. The drawing is not to scale. E F G 10. If , find the value of x. The drawing is not to scale. E F G 11. Find the value of x. (6x + 12)° 108° 12. Based on the pattern, what are the next two terms of the sequence? 2, 6, 10, 14, . . . 13. 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 2x – 9 = –3. Statement 2: x = 3 14. Use the Law of Syllogism to draw a conclusion from the two given statements. If it is Thursday, then Sara gets a paycheck. If Sara gets a paycheck, then she goes to the bank. 15. If and point F is 2/5 of the way between E and G, find the value FG. The drawing is not to scale. E F 16. What is the relationship between G and ? 1 2 m 3 4 5 6 n 7 8 17. Line r is parallel to line t. Find m 6. The diagram is not to scale. r 7 137° 1 3 t 4 2 5 6 18. Find the coordinates of the midpoint of the segment whose endpoints are H(2, 15) and K(8, 5). 19. Find the distance between points P(2, 3) and Q(9, 4) to the nearest tenth. 20. Find the image of C under the translation described by the translation rule E8 y A 4 C –8 . –4 4 –4 8 x D B –8 21. The vertices of a triangle are P(2, –8), Q(–1, 2), and R(1, 2). Name the vertices of a triangle reflected over the x axis. 22. Is a vertical compression or a vertical stretch? What coordinate rule maps ? to y A' A –8 B' 8 4 B –4 4 8 x C –4 –8 C' 23. Which two triangles similar? How do you know? 24. 25. State whether and 5 26. What is the value of x? are congruent. Justify your answer. 5 36° 21 21 xº Drawing not to scale 27. What additional information will allow you to prove the triangles congruent by the HL Theorem? A B | C | D E 28. What is the geometric mean of 81 and 4 29. What is the value of x, given that ? A x 8 P Q 32 24 B C 30. What is value of x, given that O P 9 N x Q 18 20 M ? 31. is a right triangle with altitude to the hypotenuse . What is the geometric mean of JM and KM? L J M K 32. What is the value of x? xº 58° Drawing not to scale 33. B is the midpoint of D is the midpoint of and AE = 29. Find BD. The diagram is not to scale. C B A 34. bisects D E Find the value of x. The diagram is not to scale. | | E 2x + 60 F ) 5x ) 28° D G 35. Where is the circumcenter of any given triangle? 36. Find the length of , given that is a median of the triangle and AC = 50. D A B C 37. Name the point of concurrency of the angle bisectors. 38. For a triangle, list the respective names of the points of concurrency of • perpendicular bisectors of the sides • bisectors of the angles • medians • lines containing the altitudes 39. Find the values of x, y, and z. The diagram is not to scale. 46° 18° 65° x° z° y° 40. Find the value of x. The diagram is not to scale. 43° 112° x° 41. Construct a segment bisector 42. 43. Construct a pair of parallel lines 44. Construct a line perpendicular through point F 45. 46. Describe the rule in words 47. Write a two column proof 48. Graph the reflection of the triangle over the x-axis. 49. Fall Geometry Final Review Answer Section SHORT ANSWER 1. ANS: biconditional PTS: 1 DIF: L2 REF: 2-3 Biconditionals and Definitions OBJ: 2-3.1 To write biconditionals and recognize good definitions STA: (4)(A)| (4)(B) TOP: 2-3 Problem 1 Writing a Biconditional KEY: conditional statement | biconditional statement 2. ANS: If I drink juice, then it is breakfast time. If it is breakfast time, then I drink juice. PTS: 1 DIF: L3 REF: 2-3 Biconditionals and Definitions OBJ: 2-3.1 To write biconditionals and recognize good definitions STA: (4)(A)| (4)(B) TOP: 2-3 Problem 2 Identifying the Conditionals in a Biconditional KEY: biconditional statement | conditional statement 3. ANS: The reverse is false. PTS: 1 DIF: L3 REF: 2-3 Biconditionals and Definitions OBJ: 2-3.1 To write biconditionals and recognize good definitions STA: (4)(A)| (4)(B) TOP: 2-3 Problem 3 Writing a Definition as a Biconditional KEY: biconditional statement 4. ANS: counterexample PTS: 1 DIF: L2 REF: 2-3 Biconditionals and Definitions OBJ: 2-3.1 To write biconditionals and recognize good definitions STA: (4)(A)| (4)(B) TOP: 2-3 Problem 4 Identifying Good Definitions KEY: counterexample 5. ANS: Yes, they lie on the line AC. PTS: OBJ: STA: KEY: 6. ANS: 1 DIF: L2 REF: 1-1 Points, Lines, and Planes 1-1.1 To understand basic terms and postulates of geometry (4)(A) TOP: 1-1 Problem 1 Naming Points, Lines, and Planes point | line | collinear points PTS: OBJ: STA: KEY: 1 DIF: L2 REF: 1-1 Points, Lines, and Planes 1-1.1 To understand basic terms and postulates of geometry (4)(A) TOP: 1-1 Problem 2 Naming Segments and Rays ray | opposite rays 7. ANS: a. Given b. Subtraction Property of Equality c. Multiplication Property of Equality PTS: 1 DIF: L2 REF: 2-5 Reasoning in Algebra and Geometry OBJ: 2-5.1 To connect reasoning in algebra and geometry STA: (6)(A) TOP: 2-5 Problem 3 Writing a Two-Column Proof KEY: Properties of Equality | proof | Two-column Proof 8. ANS: If you are in high school, then you are a senior; false PTS: 1 STA: G.4.B NOT: Example 3 9. ANS: 15 DIF: Level 2 REF: Geometry Sec. 2.1 KEY: conditional statement | converse | inverse | contrapositive | application PTS: 1 DIF: L2 REF: 1-2 Measuring Segments OBJ: 1-2.1 To find and compare lengths of segments STA: (2)(A) TOP: 1-2 Problem 2 Using the Segment Addition Postulate KEY: coordinate | distance 10. ANS: PTS: 1 DIF: L3 REF: 1-2 Measuring Segments OBJ: 1-2.1 To find and compare lengths of segments STA: (2)(A) TOP: 1-2 Problem 2 Using the Segment Addition Postulate KEY: coordinate | distance 11. ANS: PTS: 1 STA: G.6.A 12. ANS: 18, 22 DIF: Level 1 REF: Geometry Sec. 2.6 KEY: using angle relationships NOT: Example 4-1 PTS: 1 DIF: L3 REF: 2-1 Patterns and Conjectures OBJ: 2-1.1 To use inductive reasoning to make conjectures STA: (4)(C)| (5)(A) TOP: 2-1 Problem 1 Finding and Using a Pattern KEY: pattern | inductive reasoning 13. ANS: 2x – 9 = –3 PTS: 1 DIF: L4 REF: 2-4 Deductive Reasoning OBJ: 2-4.1 To use the Law of Detachment and the Law of Syllogism STA: (6)(A) TOP: 2-4 Problem 1 Using the Law of Detachment KEY: Law of Detachment | deductive reasoning 14. ANS: If it is Thursday, then Sara goes to the bank. PTS: 1 DIF: L3 REF: 2-4 Deductive Reasoning OBJ: 2-4.1 To use the Law of Detachment and the Law of Syllogism STA: (6)(A) TOP: 2-4 Problem 2 Using the Law of Syllogism KEY: deductive reasoning | Law of Syllogism 15. ANS: 18 PTS: 1 DIF: L4 REF: 1-2 Measuring Segments OBJ: 1-2.1 To find and compare lengths of segments STA: (2)(A) TOP: 1-2 Problem 2 Using the Segment Addition Postulate KEY: coordinate | distance | partition segment in a given ratio 16. ANS: corresponding angles PTS: OBJ: STA: KEY: 17. ANS: 43 1 DIF: L3 REF: 3-1 Lines and Angles 3-1.2 To identify angles formed by two lines and a transversal (6)(A) TOP: 3-1 Problem 3 Classifying an Angle Pair angle pairs | transversal | parallel lines PTS: 1 DIF: L3 REF: 3-2 Properties of Parallel Lines OBJ: 3-2.2 To use properties of parallel lines to find angle measures STA: (5)(A)| (6)(A) TOP: 3-2 Problem 2 Identifying Supplementary Angles KEY: parallel lines | alternate interior angles 18. ANS: (5, 10) PTS: OBJ: STA: KEY: 19. ANS: 7.1 1 DIF: L2 REF: 5-1 Midpoint and Distance in the Coordinate Plane 5-1.1 To find the midpoint of a segment by deriving and using the midpoint formula (2)(A)| (2)(B) TOP: 5-1 Problem 2 Finding the Midpoint coordinate plane | Midpoint Formula PTS: 1 DIF: L3 REF: 5-1 Midpoint and Distance in the Coordinate Plane OBJ: 5-1.2 To find the distance between two points in the coordinate plane by deriving and using the distance formula STA: (2)(A)| (2)(B) TOP: 5-1 Problem 5 Finding Distance KEY: Distance Formula | coordinate plane 20. ANS: A PTS: 1 DIF: L3 REF: 8-1 Translations OBJ: 8-1.2 To find translation images of figures STA: (3)(A)| (3)(C)| (6)(C) TOP: 8-1 Problem 3 Finding the Image of a Translation KEY: translation | preimage | image 21. ANS: PTS: 1 DIF: L3 REF: 8-2 Reflections OBJ: 8-2.1 To find reflection images of figures STA: (3)(A)| (3)(C) TOP: 8-2 Problem 1 Reflecting a Point Across a Line KEY: reflection | line of reflection 22. ANS: vertical stretch PTS: OBJ: STA: KEY: 23. ANS: 1 DIF: L3 REF: 8-8 Other Non-Rigid Transformations 8-8.2 To find compositions of transformations, including non-rigid transformations (3)(A)| (3)(B)| (3)(C) TOP: 8-8 Problem 2 Describing a Non-Rigid Transformation compression | stretch ; SAS PTS: OBJ: STA: KEY: 24. ANS: 1 DIF: L4 REF: 9-3 Proving Triangles Similar 9-3.1 To use the AA Postulate and the SAS and SSS theorems (7)(B)| (8)(A) TOP: 9-3 Problem 2 Verifying Triangle Similarity Side-Angle-Side Similarity Theorem PTS: 1 DIF: L2 REF: 4-1 Congruent Figures OBJ: 4-1.1 To recognize congruent figures and their corresponding sides and angles STA: (6)(C) TOP: 4-1 Problem 1 Finding Congruent Sides and Angles KEY: congruent polygons | corresponding parts 25. ANS: yes, by either SSS or SAS PTS: OBJ: STA: KEY: 26. ANS: 72° 1 DIF: L3 REF: 4-2 Triangle Congruence by SSS and SAS 4-2.1 To prove two triangles congruent using the SSS and SAS Postulates (5)(A)| (5)(C)| (6)(B) TOP: 4-2 Problem 4 Identifying Congruent Triangles SSS | SAS | reasoning PTS: OBJ: STA: KEY: 27. ANS: 1 DIF: L2 REF: 4-5 Isosceles and Equilateral Triangles 4-5.1 To use and apply properties of isosceles and equilateral triangles (5)(A)| (5)(C)| (6)(B)| (6)(D) TOP: 4-5 Problem 4 Using Algebra isosceles triangle | Converse of Isosceles Triangle Theorem | Triangle Angle-Sum Theorem PTS: OBJ: STA: KEY: 28. ANS: 18 1 DIF: L3 REF: 4-6 Congruence in Right Triangles 4-6.1 To prove right triangles congruent using the Hypotenuse-Leg Theorem (6)(B) TOP: 4-6 Problem 2 Writing a Proof Using the HL Theorem hypotenuse | HL Theorem | right triangle | reasoning PTS: OBJ: STA: KEY: 29. ANS: 1 DIF: L2 REF: 9-4 Similarity in Right Triangles 9-4.1 To find and use relationships in similar right triangles (7)(B)| (8)(A)| (8)(B) TOP: 9-4 Problem 3 Finding the Geometric Mean geometric mean | proportion 6 PTS: OBJ: STA: TOP: KEY: 30. ANS: x = 10 1 DIF: L2 REF: 9-5 Proportions in Triangles 9-5.1 To use the Triangle Proportionality Theorem and the Triangle-Angle-Bisector Theorem (5)(A)| (7)(B)| (8)(A) 9-5 Problem 3 Using the Triangle Proportionality Theorem Side-Splitter Theorem PTS: OBJ: STA: TOP: KEY: 31. ANS: LM 1 DIF: L3 REF: 9-5 Proportions in Triangles 9-5.1 To use the Triangle Proportionality Theorem and the Triangle-Angle-Bisector Theorem (5)(A)| (7)(B)| (8)(A) 9-5 Problem 3 Using the Triangle Proportionality Theorem Side-Splitter Theorem PTS: OBJ: STA: KEY: 32. ANS: 64° 1 DIF: L3 REF: 9-4 Similarity in Right Triangles 9-4.1 To find and use relationships in similar right triangles (7)(B)| (8)(A)| (8)(B) TOP: 9-4 Problem 4 Identifying Proportions in a Right Triangle geometric mean PTS: OBJ: STA: KEY: 33. ANS: 14.5 1 DIF: L2 REF: 4-5 Isosceles and Equilateral Triangles 4-5.1 To use and apply properties of isosceles and equilateral triangles (5)(A)| (5)(C)| (6)(B)| (6)(D) TOP: 4-5 Problem 4 Using Algebra isosceles triangle | Isosceles Triangle Theorem | Triangle Angle-Sum Theorem | word problem PTS: OBJ: STA: KEY: 34. ANS: 20 1 DIF: L2 REF: 5-2 Midsegments of Triangles 5-2.1 To use properties of midsegments to solve problems (6)(D) TOP: 5-2 Problem 3 Finding Lengths midpoint | midsegment | Triangle Midsegment Theorem PTS: 1 DIF: L3 REF: 5-3 Perpendicular and Angle Bisectors OBJ: 5-3.1 To use properties of perpendicular bisectors and angle bisectors STA: (5)(C)| (6)(A) TOP: 5-3 Problem 5 Using the Angle Bisector Theorem KEY: Angle Bisector Theorem | angle bisector 35. ANS: the point of concurrency of the perpendicular bisectors of the sides of the triangle PTS: OBJ: STA: KEY: 1 DIF: L2 REF: 5-4 Bisectors in Triangles 5-4.1 To identify properties of perpendicular bisectors and angle bisectors (5)(A)| (5)(C)| (6)(D) TOP: 5-4 Problem 2 Finding the Circumcenter of a Triangle point of concurrency | concurrent | circumcenter of a triangle 36. ANS: 25 PTS: OBJ: STA: KEY: 37. ANS: C 1 DIF: L2 REF: 5-5 Medians and Altitudes 5-5.1 To identify properties of medians and altitudes of a triangle (6)(D) TOP: 5-5 Problem 1 Finding the Length of a Median median of a triangle PTS: 1 DIF: L3 REF: 5-4 Bisectors in Triangles OBJ: 5-4.1 To identify properties of perpendicular bisectors and angle bisectors STA: (5)(A)| (5)(C)| (6)(D) TOP: 5-4 Problem 4 Identifying and Using the Incenter of a Triangle KEY: angle bisector | incenter of a triangle | point of concurrency 38. ANS: circumcenter incenter centroid orthocenter PTS: 1 DIF: L3 REF: 5-5 Medians and Altitudes OBJ: 5-5.1 To identify properties of medians and altitudes of a triangle STA: (6)(D) TOP: 5-5 Problem 3 Finding the Orthocenter KEY: angle bisector | circumcenter of a triangle | centroid of a triangle | orthocenter of a triangle | median | altitude of a triangle | perpendicular bisector 39. ANS: PTS: 1 DIF: L3 REF: 3-5 Parallel Lines and Triangles OBJ: 3-5.1 To find measures of angles of triangles STA: (6)(D) TOP: 3-5 Problem 2 Using the Triangle Angle-Sum Theorem KEY: triangle | sum of angles of a triangle 40. ANS: 69 PTS: OBJ: TOP: KEY: 41. ANS: 1 DIF: L2 REF: 3-5 Parallel Lines and Triangles 3-5.1 To find measures of angles of triangles STA: (6)(D) 3-5 Problem 3 Using the Triangle Exterior Angle Theorem triangle | sum of angles of a triangle | exterior angle of a polygon | remote interior angles PTS: 1 42. ANS: y+9=3(x+3) PTS: 1 43. ANS: PTS: 1 44. ANS: PTS: 1 45. ANS: PTS: 1 46. ANS: PTS: 1 47. ANS: PTS: 1 48. ANS: graph PTS: 1 49. ANS: Angle C PTS: 1