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Name: _____________________________________ Date: _______________ Block: __________ CP GEOMETRY Final Exam Study Packet *Figures not drawn to scale unless a scale is indicated* Formulas you will be given on the exam: Chapter 1: Foundations of Geometry Collinear Coplanar Angle bisector: Segment bisector (midpoint): Complementary: Supplementary: Adjacent angles: Linear pair: Vertical angles: Segment Addition Postulate: Angle Addition Postulate: Points on the same line Points on the same plane A ray that cuts an angle into two congruent halves. A point that cuts a segment into two congruent halves. Add up to 90° Add up to 180° Angles that share a side and do not overlap A pair of angles that are adjacent and supplementary Non-adjacent angles formed by two intersecting lines, they are congruent The length of a whole segment is equal to the sum of its parts’ lengths The measure of a whole angle is equal to the sum of its parts’ measures Distance Formula: d ( x2 x1 ) 2 Midpoint of a Segment: x1 x2 y1 , 1. 2 ( y2 y2 2 y1 ) 2 or plot the points and use Pythagorean Theorem (a2 + b2 = c2) or find average of the x’s and of the y’s is the angle bisector of CAD. m CAB = (6x + 8)°, and m CAD = (15x + 2)°. What is the value of x? A. B. 3. intersects at E. If is the perpendicular bisector of , which of the following MUST be true? 2 3 A. 80 21 C. B. D. 14 C. 3 170 D. 21 2. What is the midpoint of the segment whose endpoints are (6, 8) and (4, 0)? A. (5, 4) 4. m A = (4x + 6)°, and its supplement has measure (6x – 16)°. What is the value of x? A. 10 B. 11 C. 17 D. 19 B. (1, 3) C. (3, 6) D. (3, 2) 5. What is the intersection of A. H B. F C. D. and ? Chapter 2: Geometric Reasoning conditional converse inverse contrapositive original pq reverse order qp put “not” in front of both ~p ~q reverse order AND put “not” in front of both ~q ~p “If I have gotten a good grade, then I am smiling.” “If I am smiling, then I have gotten a good grade.” “If I have not gotten a good grade, then I am not smiling.” “If I am not smiling, then I have not gotten a good grade.” Biconditional Statement: Conjecture: Counterexample: “If and only if” (combines a statement and its converse) A statement based on observation An example that proves a conjecture or statement wrong. Law of Detachment: If p q is a valid conditional statement, and p is true, then q is true. (“If I am happy, then I am smiling.” I am happy. Therefore, I am smiling.) If p q and q r are valid conditional statements, then p r is a valid statement. (“If I have gotten a good grade, then I am happy. If I am happy, then I am smiling.” So “If I have gotten a good grade, then I am smiling” is a valid statement.) Law of Syllogism: Addition Property of Equality Subtraction Property of Equality Multiplication Property of Equality Division Property of Equality We can add the same thing to both sides of an equation. We can subtract the same thing from both sides of an equation. We can multiply both sides of an equation by the same thing. We can divide both sides of an equation by the same thing (except 0). Reflexive Property of Equality Symmetric Property of Equality Transitive Property of Equality x=x If x = 5 , then 5 = x. If x = y and y = 5, then x = 5. Substitution Property Distributive Property of Equality Commutative Property Associative Property If x = 5, then we can “plug in” 5 for x anywhere we want to. We can distribute things outside the parentheses to inside: 5(x + 2) = 5x + 10 We can add or multiply in any order: 3 + 5 = 5 + 3 We can move parentheses around: 3 + (5 + 1) = (3 + 5) + 1 Reflexive Property of Congruence Symmetric Property of Congruence Transitive Property of Congruence Definition of Congruence A A If A B, then If A B and When you go from B A. B C, then A to = or = to . C. 6. “All Blingots are Whozits.” Based on this statement, which of the following must be a valid statement? A. If Carl is a Whozit, then he is a Blingot. B. If Carl is not a Blingot, then he is not a Whozit. C. If Carl is not a Whozit, then he is not a Blingot. 10. Fill in the proof below. Given: A and B are complementary. m A = 45° Prove: m A = m B A and B are complementary. m A + m B = 90° 45° + m B = 90° D. None of the above. m B = 45° 7. Which of the following is the converse of the statement, “If an angle is acute, then its measure is between 0 and 90 degrees”? m A=m B Given Definition of Complementary Substitution Property of Equality _____________________ Transitive Property of Equality A. “If an angle’s measure is between 0 and 90 degrees, then it is acute.” A. Definition of Complementary B. “If an angle’s measure is not between 0 and 90 degrees, then it is not acute.” C. Subtraction Property of Equality C. “If an angle is not acute, then its measure is not between 0 and 90 degrees” D. None of the above. B. Definition of Congruence D. Transitive Property of Equality 11. AB = CD. Which property says that CD = AB? A. Reflexive Property of Equality 8. “If something is a Bork, then it is either Glubous or Blatious.” Based on this statement, which of the following must be a valid statement? A. If something is not a Bork, then it is neither Glubous nor Blatious. B. If something is Glubous, then it is a Bork. C. If something is not Glubous, then it is not a Bork. D. None of the above. 9. “If a Planzer has a Boon, then the Planzer is either Yoppy or Flinky.” B. Subtraction Property of Equality C. Symmetric Property of Equality D. Transitive Property of Equality 12. What can you deduce from the statements: “If you pass the final exam, then you will pass the class. If you pass the class, then you will make your parents happy.” A. If you make your parents happy, then you will pass the class. B. If you pass the class, then you will make your parents happy. Which of the following could serve as a counterexample to the statement above? C. If you do not pass the final exam, then you will not pass the class. A. A Planzer that is not Yoppy. D. If you pass the final exam, then you will make your parents happy. B. A Planzer that does not have a Boon. C. Something Yoppy that is not a Planzer. D. A Planzer with a Boon that is neither Yoppy nor Flinky. Chapter 3: Parallel and Perpendicular Lines When parallel lines are cut by a transversal, then … Vertical s are congruent Corresponding s are congruent // Parallel Lines: Perpendicular Lines: Skew Lines: Slope of a line: Alternate Interior are congruent s Alternate Exterior are congruent 15. In the diagram below, l || m || n. If m 2 = 84°, what is m 12? 1 3 (19b)° (16b – 10)° 9 11 n C. B. 10 3 D. 38 7 A. 84° 190 3 C. 104° 5 6 7 8 m n 16. What is the slope of a line perpendicular to a line going through the points (0, -8) and (2, 4)? 2 4 10 12 D. 106° j 1 l B. 96° 14. In the diagram below, which of the following pairs of angles are alternate interior angles? 3 2 4 5 6 7 8 m 20 7 Consecutive Interior s are supplementary Coplanar lines that never intersect, slopes are the same Coplanar lines that intersect to form right angles, slopes are opposite reciprocals Noncoplanar lines that never intersect y 2 - y1 rise or x 2 - x1 run 13. In the diagram below, m || n. What is the value of b? A. s 1 6 m A. n C. -6 A. 3 and 5 B. 3 and 4 C. 3 and 6 D. 1 and 8 B. 6 D. 1 6 Chapters 4 and 5: Triangles Scalene triangle: Isosceles triangle: Equilateral triangle: no congruent sides at least two congruent sides three congruent sides Acute triangle: Right triangle: Obtuse triangle: Equiangular triangle: all angles are acute one angle is right one angle is obtuse all angles are congruent (60°) Triangle Sum Theorem: The three angles of a triangle add up to 180°. Exterior Angle Theorem: An exterior angle of a triangle is equal to the sum of the two remote, interior angles. Third Angles Theorem: If 2 angles in a triangle are congruent to 2 angles in another triangle, 3 rd angles are congruent. Base angles of an isosceles triangle: the congruent angles (opposite the congruent sides) SSS CPCTC: SAS ASA AAS HL Corresponding Parts of Congruent Triangles are Congruent Median a segment whose endpoints are a vertex and midpoint of a triangle. Medians meet at the Centroid (balancing point, 1:2 ratio) Altitude a perpendicular segment from a vertex to the opposite side of a triangle. Altitudes meet at the Orthocenter Perpendicular Bisector a segment that is perpendicular to another segment at its midpoint. Perpendicular bisectors meet at the Circumcenter (equidistant to vertices) Angle Bisector a segment that divides an angle in half. Angle bisectors meet at the Incenter (equidistant to sides) If a point is on the perpendicular bisector of a segment, then it is equidistant to the endpoints of the segment. If a point is on the angle bisector of a segment, then it is equidistant to the sides of the angle. Triangle Inequality Theorem: short + short > long (makes a triangle) The largest angle of a triangle is opposite its longest side. Likewise, the smallest angle is opposite the shortest side. Midsegment of a Triangle: Segment connecting the midpoints of the sides. Half the length of the side it is parallel to. 17. Which of the following would be enough to prove ΔABC ΔDEF by SAS? A. BC EF , AC DF , A D B. AB DE , BC EF , A D C. AB DE , AC DF , B E D. AB DE , AC DF , A D 18. Based on the drawing below, what is true? 19 C A 12 9 20. Given that ΔABC ΔDCB, which of the following is NOT necessarily true? A B C D A. AC || BD B. AC BD C. m CBD = m CBA D. m BCA = m CBD B A. m A < m B < m C B. m A < m C < m B 21. Which of the following could be the side lengths of a triangle? C. m B < m C < m A A. 2, 5, 3 D. m B < m A < m C B. 5, 13, 7 C. 8, 4, 4 D. 4, 11, 8 19. Based on the diagram below, which of the following must be true? 22. Find the value of x so that HL . MNO D C B A A. m CAD = 45° A. 5 B. 11 B. AB BD C. 4 D. 7 C. DB DC D. CDB is a right angle. PRO by Chapter 6: Polygons and Quadrilaterals triangle 3 sides quadrilateral 4 sides pentagon 5 sides hexagon 6 sides heptagon 7 sides octagon 8 sides nonagon 9 sides decagon 10 sides The sum of the measures of the interior angles of a polygon with n sides is (n – 2) • 180°. The sum of the measures of the exterior angles of a polygon with n sides is 360°. Quadrilateral: Parallelogram: 4 sides 2 pairs of parallel sides ( opposite sides are congruent, opposite angles are congruent, consecutive angles are supplementary, diagonals bisect each other) Rectangle: Parallelogram with four congruent (right) angles ( diagonals are congruent) Rhombus: Parallelogram with four congruent sides ( diagonals are perpendicular, diagonals bisect angles) Square: A rectangle and a rhombus Trapezoid: Exactly 1 pair of parallel sides (midsegment is average of the bases and parallel to bases) Isosceles Trapezoid: Trapezoid with congruent legs ( diagonals are congruent, base angles are congruent) Kite: 2 pairs of consecutive congruent sides ( diagonals are perpendicular, one pair opposite angles congruent) 23. What is the measure of each interior angle of a regular octagon? A. 45° 26. Refer to rhombus FISH. If m FIX = 18°, what is m FIS? F I X B. 135° C. 180° H D. 1,080° S A. 18° 24. What is m A. 30° NMR in the square MNOP? N M B. 90° C. 36° D. 6° C. 45° D. 60° B. 9° R P O 27. OL is the midsegment. If CA = 16 and OL = 12, find RM. 25. Which of the following are properties of a rhombus? A C I. Diagonals are congruent L O II. Diagonals are perpendicular III. Diagonals bisect each other. A. 28 A. I and II B. I and III B. 4 C. II and III D. I, II, and III C. 2 D. 8 R M Chapter 7: Similarity scale factor = new old The angles are congruent; the sides are proportional. AA Similarity SSS Similarity SAS Similarity Ratio of perimeters of similar figures is = scale factor Ratio of areas of similar figures is = (scale factor)2 A line is parallel to a side of a triangle if and only if it divides the sides it intersects proportionally. An angle bisector divides the opposite side proportionally to the other two sides. 28. What is the value of x in the figure below? 6 A. 25 64 B. 5 8 4 x 8 A. 3 B. 4 C. 30. If the scale factor of two similar figures is 5 : 8, what is the ratio of their perimeters? C. 10 4 D. 10 2 16 3 D. 12 29. The two polygons below are similar as they appear. What is the value of x? 9 6 6 31. The two trapezoids below are similar. To the nearest tenth, what is the value of x? 10 x x 6 A. 2 3 B. 4 C. 6 D. 9 A. 2.1 B. 2.9 C. 4.2 D. 6 8 Chapter 8: Right Triangles & Trigonometry Pythagorean Theorem: a2 + b 2 = c 2 Must be a right triangle, a and b are legs, c is hypotenuse Pythagorean Triple: 3 numbers that satisfy the Pythagorean Theorem example: 3, 4, 5 and 5, 12, 13 Similar Right Triangles: Geometric Mean = product Special Right Triangles: 45 30 x 2 2x x 3 x 45 60 x Trigonometry: SohCahToa sin x opposite hypotenuse 32. What is the value of x in the figure below? 1 x 3 x cos x adjacent hypotenuse tan x opposite adjacent 34. The angle of elevation of a rope tied from a stake in the ground to the top of a 3-foot-tall fence is 72°. To the nearest tenth of a foot, how far is the stake in the ground from the base of the fence? sin 72° = 0.951 cos 72° = 0.309 tan 72° = 3.078 A. 1 foot B. 3.2 feet A. 2 2 C. 6.4 feet B. 8 D. 9.7 feet C. 6 2 35. Which of the following represents the value of x? D. 9 33. Find the value of x to the nearest tenth. A. 14 sin 38° B. 14 cos 38° A. 57.4 B. 28.3 C. 0.5 D. 32.6 7 13 C. x D. x 14 14 sin 38 14 cos 38 38 Chapter 10: Area, Perimeter, and Circumference Area of parallelogram, rectangle, and square = bh b b dd Area of a trapezoid = 1 2 h Area of a rhombus and kite = 1 2 2 2 bh Area of a triangle = Area of a circle = πr2 Circumference of a circle =2πr 2 36. For what value of x will the area of the trapezoid below be 48 square units? 39. What is the area of the shaded region below? 4 x 8 A. 4 B. 8 A. 8 m 2 C. 6 D. 12 B. 64 m 2 C. 16 m 2 37. What is the height of a triangle whose area is 50 cm2 and base is 20 cm? A. 30 cm B. 5 cm C. 10 cm D. 12.5 cm D. 32 m 2 40. What is the area of the trapezoid below? 8 38. What is the area of a kite with d1 = (x + 2) and d2 = (2x + 4)? 6 60 A. 2x2 + 8x + 8 B. x2 + 4x + 4 C. 3x + 6 D. 1.5x + 3 A. 24 square units B. 48 square units C. 33 3 square units D. 66 3 square units Chapter 11: Surface Area and Volume Length is measured in units (cm) Area is measured in square units (cm2) Volume is measured in cubic units (cm3) Ratio of SA of similar solids= (scale factor)2 Ratio of Volumes of similar solids = (scale factor)3 41. What is the volume of the prism below? 44. What is the surface area of the right cylinder below? A. 18π cm2 B. 36π cm2 6 cm D. 15π cm2 9 cm 4 cm A. 36 5 cm3 B. 108 cm 6 cm C. 54π cm2 3 cm 45. What is the radius of a sphere with surface area 12π cm2? 3 3 cm C. 72 5 cm3 A. D. 216 cm3 B. 3 cm C. 9 cm 42. The figure below shows a right square pyramid placed on top of a cube. What is the surface area of the resulting solid? A. 450 in2 7 in D. 4 3 cm 46. The solids below are similar. Which of the following is the ratio of their surface areas? B. 531 in2 C. 567 in2 D. 918 in2 9 in 43. What is the volume of a sphere with diameter 12 feet? A. 288π ft3 15 cm A. B. 5 4 C. 25 16 D. 125 64 B. 864π ft3 C. 1152π ft3 5 2 D. 2304π ft3 12 cm Transformations translation rotation 47. The segment below is reflected in the x-axis. What are the coordinates of A’? reflection dilation 49. What is the image of the point (4, -5) if it is translated by (x – 3, y + 2)? A. (7, -3) B. (1, -7) C. (1, 3) D. (1, -3) A. (1, -2) 50. What is the image of the point (0, -3) if it is reflected over the y-axis? B. (-1, -2) C. (1, 2) A. (-3, 0) D. (-2, 1) B. (3, 0) C. (-3, 0) 48. What type of transformation turns the dashed figure into the solid figure? A. dilation B. reflection C. rotation D. translation D. (0, -3)