Honors Geometry KEY Review Exercises for the January Exam

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1 Honors Geometry KEY Review Exercises for the January Exam Here is a miscellany of exercises to help you prepare for the semester examination. You should also use your class notes, homework, quizzes, and tests for more exercises. Write one of the words SOMETIMES, ALWAYS, or NEVER to complete each statement. You need to think of all possibilities to decide which word is correct. 1. The exterior angles of a hexagon always sum to Three points are sometimes collinear. 3. Three points are always coplanar. 4. Parallel lines are always coplanar. 5. Through two points, there is always exactly one line. 6. If two rays share a common endpoint, they are sometimes opposite rays. 7. A right triangle always has a pair of complementary interior angles. 8. If two angles of a quadrilateral have measure and 80 0, then the other two interior angles are never complementary. (Note: The other two angles, in this case, are always supplementary.) TRUE or FALSE? 9. If two lines are perpendicular to the same line, then they are parallel to each other. (FALSE: If they are non-coplanar, they are not necessarily parallel. If all lines are coplanar, this is a true statement.) 10. In an orthographic drawing of a solid made of cubes, the right view and the left view are identical. (FALSE. Ignore this question!) 11. If parallel lines are cut by a transversal, then alternate interior angles could be complementary. (TRUE: The importance of the word could is that it is not as strong as IS. This does not make the statement false.) 12. The supplements of congruent angles are congruent. (TRUE) 13. Perpendicular is a symmetric relation. (TRUE: This says that if a b, then b a) 14. Greater than is transitive. (TRUE) 15. Supplementary angles must be adjacent. (FALSE)

2 Honors Geometry Exam Review Questions page Each interior angle of a regular hexagon has a measure of (TRUE, using ( n 2)180 with n = 6.) n Complete the following. 17. An angle is the union of two rays with a common endpoint. 18. Perpendicular lines form right angles. 19. If parallel lines are cut by a transversal, then same side interior angles are supplementary. (Another answer is same side exterior angles ) 20. In a conditional statement, if you reverse the hypothesis and conclusion, you get the converse of the original statement. 21. If an original conditional statement is false, then the contrapositive of the original statement must also be false. 22. Examine the diagram. True or False a) A, B, D, and E are coplanar true b) B and C are collinear true F c) A, F, and C are collinear true d) F lies in plane P false A P e) Q FC intersects plane P at point A and plane Q at point C true C B D E Name the following: f) The intersection of planes P and Q BD g) A pair of skew lines BD and FC h) The intersection of plane Q and FC C 23. Refer to the diagram below. P R a) If m PXR = 42,find the measure of these angles: m RXT m QXS m SXP X T b) If m SPX = 22 and PX = SX, find m PXS. = = S Q c) If RPX SQX, then what lines must be parallel? Give the reason. SQ PR b/c AIC P d) If SXQ TQX, then what lines must be parallel? Give the reason. SX TQ b/c AIC P

3 Honors Geometry Exam Review Questions page Write three postulates. Some samples are provided. There are others. a. Through two points, there is one line. b. If two parallel lines are cut by a transversal, then corresponding angles are congruent. c. If two planes intersect, then their intersection is a line. 25. Write three theorems. a. In a triangle, the sum of the interior angles is 180 degrees. b. If two lines are cut by a transversal, then the alternate interior angles are congruent. c. Vertical angles are congruent. Solve for any variable in the drawings

4 Honors Geometry Exam Review Questions page gon 33. each interior angle = 34. M is the midpoint of CD. The coordinates are: C (6, x) M (1, x 4) D (-4, 11) 35. Find the length of CD if C (6, -3) and ( 0, 8). 36. Where is the midpoint of AQ if A(9, 4) and Q(-1, 11)? 37. A line passes through the points A( -4, 5) and B(2, -7). a) Find AB = 2 2 (5 7) ( 4 2) b) Find the slope of AB = c) Find an equation of AB. Write your answer in point-slope form. y + 7 = -2(x 2) OR y 5 = -2(x + 4) d) Find the coordinate of the midpoint of AB M, 2 2 = (-1,-1)

5 Honors Geometry Exam Review Questions page 5 Construct these with compass and straightedge. These are done approximately, since I do not have compass and straightedge that works on the tablet. 38. an isosceles right triangle with sides (non-hypotenuse) given. 39. a line perpendicular to AB through C. 40. a square with side EF. 41. a 45 0 angle 42. the complement of < A

6 Honors Geometry Exam Review Questions page 6 Draw each of these. 43. This isometric drawing has a volume of 9 cubes. Make a drawing with cubes whose volume is 13 cubes. \ 44. Calculate x. 45. Calculate x. x = 180 x + x + 26 = 180 x = 34 2x = 154 x = Write Pizza is served if and only if today is Tuesday as two if then statements. If pizza is served, then today is Tuesday. and If today is Tuesday, then pizza is served. Write the only if half as an if then statement. If pizza is served, then today is Tuesday

7 Honors Geometry Exam Review Questions page Consider a set on n noncollinear points. Connect each point with all other points. Count the number of line segments. # of segments = 3 # of segments = 6 # of segments = 10 If there are n noncollinear points, what is an expression for the total number of segments which can be drawn? f(n) = n + nn ( 3) 2 This is the number of sides of the polygon (n) plus the number of diagonals drawn from each vertex times the number of vertices. The number of diagonals is divided by two, since they were all counted twice. 48. How many sides and diagonals (total) does a decagon have? a 27-gon? an n-gon? From above, a decagon has (10)(7) = 45. A 27-gon has (27)(24) = An n-gon has n + nn ( 3) 2 sides and diagonals total. How is this related to the handshake problem? It is the same question as the handshake problem. 49. Given: A rolling stone gathers no moss. a) Rewrite the statement as a conditional. If a stone is rolling, then it gathers no moss. b) Write the converse. If a stone gathers no moss, then it is rolling. c) Write a biconditional combining the statement and its converse.

8 Honors Geometry Exam Review Questions page 8 A stone is rolling if and only if it gathers no moss. d) Assume the given statement is true. Which of these statements must be true? This big gray stone is not rolling, therefore it is gathering moss. (This is using the inverse, whose truth is not known from the truth of the original.) That red stone is gathering moss, therefore it is not rolling. TRUE (This is using the contrapositive, which is TRUE if the original is true.) That green stone is not gathering moss, therefore it is rolling. (This is using the converse, so we do not know whether it is true or false.) 50. Make a truth table for this logic statement: ( P Q) ( Q ~ P) P Q T T T F F T F F 51. Make a conclusion (if possible) and name the pattern of reasoning. P R ~ R W ( S R) W P R a) R P b) R W c) ( S R) W d) R Q P Q modus tollens law of disjunction modus ponens law of syllogism (law of (law of contrapositive) detachment) 52. Assuming A and B are both true, P and Q are both false, and X and Y are of unknown truth value. Determine (if possible) the truth value of each statement. a) A ( B X) Since B is true, then B X is true (regardless of the truth value of X). So with A being true, A ( B X) is true AND true, so the entire statement is TRUE. b) B ( P A)

9 Honors Geometry Exam Review Questions page 9 Starting with P A, this conjoins a FALSE and TRUE, which is TRUE. Then the if then gives TRUE TRUE, so the entire statement is TRUE. c) ( A X) ( B Q) A X is TRUE because A is TRUE. B Q is FALSE because Q is FALSE. The conditional is FALSE because TRUE FALSE leads to a false statement. 53. Write a logical expression which is logically equivalent to ( P Q) Q. P Q P Q Q ( P Q) Q T T T F F T F T T T F T T F F F F F T T This is logically equivalent to Q 54. Write the inverse of People wear flip flops only if they hate real shoes. First, re-write this statement in an if then form. If people wear flip flops, then they hate real shoes. The inverse is: If people do not wear flip flops, then they do not hate real shoes. 55. Give an example of a syllogism. If A B and B C, then A C 56. Give an example of the law of disjunction. If (A or B) and not B, then we can conclude A. 6 d 57. Which lines (if any) must be parallel? a a) 1 2 a b b) 6 7 c d c) 9 4 a b d) a b e) 11 8 none f) 5 and 11 are supp a b g) 4 and 12 are supp c d h) 5 and 10 are supp none b 2 12 c

10 Honors Geometry Exam Review Questions page Assume that a b and c d m< 1 = 96 0 m < 6 = 40 0 Calculate all of the other numbered angles. b a d 9 c Draw each of these triangles (if possible). a) right triangle c) isosceles acute triangle b) scalene obtuse triangle 60. Draw two horizontal planes, A and B, a vertical plane, C, and a line DE so that D lies in C, and E lies in A

11 Honors Geometry Exam Review Questions page What is the locus of the center of a sphere that rolls around on a rectangular table top whose surface measures 5 feet by 3 feet? The locus is a part of a plane which is 5 feet by 3 feet (a rectangle) which is above the table top by the size of the radius of the sphere. 62. What are the loci of points that are equidistant from two points, A and B and also equidistant from two parallel planes? The loci is the intersection of a plane which is equidistant from A and B and the perpendicular bisector of AB, and a plane which is between the parallel planes. This intersection is either a line, a plane, or the null set (no intersection). 63. a) What is the locus of points, in a plane, which are equidistant from two perpendicular rays which have a common endpoint? The locus is a line which is the bisector of the angle created by the two rays, so the Bisector forms a 45 0 angle with the rays. b) What is the locus of points which are equidistant from two perpendicular rays which have a common endpoint? The locus is a plane which is the bisector of the angle created by the two rays, so the Bisector forms a 45 0 angle with the rays.

12 Calculate the angles x = (x + 20) x = x + 54 x = 54 Angles measure 108 and The angles of a triangle are in the ratio 2:3:5. Find the measure of the largest angle in the triangle. Always support your answers with appropriate algebra. Use 2x, 3x, and 5x 2x + 3x + 5x = 180 so 10x = 180, thus x = 18 The angles measure 36, 54, and 90. So 90 is the largest angle in the triangle. 66. Y = = 154 x +y = 180, so x = 26 x 10 = 16 z + (x 10) = x So z + 16 = 26, making z = 10 w + z = 180, so w = x = 180 x = 88

13 Geometry Exam Review Exercises page Find the measure of each interior angle of a regular octagon. ( n 2)180 (8 2) n How many sides does a regular polygon have if each exterior angle measures 20 0? n = 360 n = 18 sides n 70. An interior angle and an exterior angle of a regular polygon differ by How large is the interior angle? ( n 2) n = 100n n n 180n 100n = n = 720 n = 9 sides So the interior angles are (Supp of the 40 0 exterior angles). 71. Find the sum of the measures of the interior angles of a dodecagon. (n 2)180 (12 2) How many sides does a regular polygon have if the interior angles are 20 0 more than 3 times the exterior angles? 180( n 2) (n-2) = 3* n n n 180n 360 = 3* n 160n = 1440 n = What is the sum of the exterior angles in a 64-gon? (Enough said!)

14 Geometry Exam Review Exercises page 14 Proofs. 74. Given: BA BC, < 1 < 3 Prove: <2 is comp to < 3 Proof: 1. BA BC 1. given 2. < 1 is comp to < 2 2. Perpendicular pairs are comp. 3. < 1 < 3 3. given 4. < 2 is comp to < 3 4. If an angle is comp to one of two congruent angles, then it is comp to the other angle also. 75. Given: a b, < 1 < 3 Prove: c d Proof: 1. a b 1. given 2. < 2 < 3 2. P AEC 3. < 1 < 3 3. given 4. < 1 < 2 4. Transitive for 5. c d 5. AIC P

15 Geometry Exam Review Exercises page Given: < 1 < 4 Prove: < 2 < 3 Proof: 1. < 1 is supp to < 2 1. Linear pairs are supp 2. < 3 is supp to < 4 2. Linear pairs are supp 3. < 1 < 4 3. Given 4. < 2 < 3 4. Congruent supps thm (If two < s are supp to the same < or < s, then the < s are ) 77. Given: < DAB < DBA < 2 < 3 Prove: < 1 < 4 Proof: 1. < DAB < DBA 1. Given 2. m< DAB = m< DBA 2. Defn of < s 3. m < 1 + m < 2 = m < DAB 3. Angle addition post. 4. m < 3 + m < 4 = m < DBA 4. Angle addition post 5. m < 1 + m < 2 = m < 3 + m < 4 5. Substitution 6. < 2 < 3 6. given 7. m < 2 = m < 3 7. Defn of < s 8. m < 1 = m < 4 8. Subtraction prop of =

16 Geometry Exam Review Exercises page < 1 < 4 9. Defn of < s Calculate the areas and volumes Area = 1 2 bh 1 (8)(5) = 20 2 area = bh. By the Pythagorean Theorem, the base is 12. So area = (12)(5) = Area = bh = (12)6) = 72 area =(8)(6) ( ) =18.5

17 Geometry Exam Review Exercises page A figure is drawn on lattice paper (dot paper) with 4 interior points and 11 points on its boundary. What is its area. Pick s Theorem says that the area equals 1 2 (boundary points) + interior points 1, so the area is.5(11)+ 4-1 = 8.5 Area = (6)(6) + 1 (6)(9) = = The area of a trapezoid is A = 1 ( ) hb b Show why this formulas should be true. (i.e., drive the formula) Area = 1 2 xh + b 1H h(b 2 x b 1 ) = 1 2 xh (2b 1h) hb xh b 1h = 1 2 h(x + 2b 1h + hb 2 xh b 1 h) = 1 2 (b 1 + b 2 )