Honors Geometry KEY Review Exercises for the December Exam

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1 Honors Geometry KEY Review Exercises for the December 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 converse of a conditional sometimes has the same truth value as the original conditional. 2. 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. The locus of points which are 6 inches from point, A, sometimes is a circle with radius If same side interior angles formed from two lines and a transversal measure and 80 0, then the two lines are always parallel. 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) 11. If parallel lines are cut by a transversal, then alternate interior angles could be complementary. (FALSE: The importance of the word could is that it is not as strong as IS.) 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 (Ignore this question. 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 Make a drawing which clearly shows 3 skew lines, and one line parallel to one of the skew lines. 33. The vertex of an angle is at (3, 7). A(-1,5) is on one ray of the angle, and B (6,9) is on the other ray. Calculate the measure of the angle. (Check in the Coordinate Geometry packet for the method.) 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 =

5 Honors Geometry Exam Review Questions page 5 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) 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

6 Honors Geometry Exam Review Questions page the complement of < A Draw each of these. 43. This isometric drawing has a volume of 9 cubes. Make a drawing with cubes whose volume is 13 cubes. \

7 Honors Geometry Exam Review Questions page What are the loci of points which are equidistant from two points A, and B, and which are 9 inches from a point C? The separate locus answers are: (1) a plane which is the perpendicular bisector of AB. and (2) a sphere with radius 9 and center C. The loci (the intersections) are points, 1 point, or a circle. 45. If a double the size of an angle (say from 20 0 to 40 0 ), what happens to the supplement of the complement of the angle? The supp of the comp of x = 180 (90 x) which simplifies to 90 + x The supp of the comp of 2x = 180 (90 2x) which simplifies to x So the result is larger by x 46. 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 47. 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?

8 Honors Geometry Exam Review Questions page 8 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. This is related to the handshake problem, too. 48. 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. 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.) 49. Make a truth table for this logic statement: ( P Q) ( Q ~ P) P Q T T T F F T F F

9 Honors Geometry Exam Review Questions page 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) 51. 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) 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. 52. 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 53. 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.

10 Honors Geometry Exam Review Questions page 10 The inverse is: If people do not wear flip flops, then they do not hate real shoes. 54. Give an example of a syllogism. If A B and B C, then A C 55. Give an example of the law of disjunction. If (A or B) and not B, then we can conclude A. 56. 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. 57. Which lines (if any) must be parallel? 6 d 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 a Assume that a b and c d m< 1 = 96 0 m < 6 = 40 0 c 7 Calculate all of the other numbered angles that we can to date.

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. 60. 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). 61. 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 Proofs. 62. 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 63. 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

13 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 ) 65. 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 = 9. < 1 < 4 9. Defn of < s

14 Geometry Exam Review Exercises page 14 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

15 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 )

Honors Geometry KEY Review Exercises for the January Exam

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