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2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 Name_________________________________ FCS, Mr. Garcia Date___________________ Period______ This is your semester 1 exam review study guide. It is designed for you to do a portion each day until the day of the exam. You may use the following formula to calculate your semester grade given an assumed final exam grade. (Current Re nWeb Grade)x.90 + (semester exam) x .10 = final grade Topics that we have covered on chapters 1 through 4 are outlined below for your review. =============================================================================== Chapter 1: Points, Lines, and Planes Point, line, plane – are undefined terms. They do not need to be defined. Definitions or defined terms are explained using undefined terms and/or other defined tersms. Space is defined as a boundless, 3-dimensional set of all points. Space can contain lines and planes. 1. How do you name a line? ______, a line segment? _______, a ray? ________, a plane? __________, 2. an angle, ____________, a triangle ? ________, a quadrilateral? _________, a pentagon? _________. 3. What does it mean for 3 or more points to be collinear? _______________________________, Noncollinear? ____________________________________. 4. What does it mean for 3 or more points to be coplanar? _______________________________, Noncoplanar? _______________________________. Relationships of lines and planes: 5. What does it mean for 2 lines to be parallel? ______________________________________. 6. What is the symbol for parallel? ______ What is the symbol for perpendicular? ______. 7. When 2 lines intersect, they intersect at a p________________. 8. When a line and a plane intersect, they intersect at a p____________________. 9. When 2 planes intersect, they intersect at a l __________________________. 10. An angle bisector could be a s_________________, a l____________________, or a r____________. 11. Any segment, line, or plane that intersects a segnment at its m_________________ is called a Segment b______________________. 12. When a line segment, a ray, or a line, bisects a segment, the bisector creates two s__________________ that are equal in m____________________, or equal in l________________. 13. If 2 segments are equal in length, or in measure, then they are said to be c____________________. The postulate that states this is called? _____________________. Look it up in you textbook. 14. When a line segment, a ray, or a line, bisects an angle, it creates two c__________________ angles, and their measures are e_________________. 15. What is the difference between an expression and an equation? Write an example of each. 16. A point where the 2 sides of an angle meet is called? v____________. 1 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia y 17. The slope of a line can be calculated when you are given 2 p________________. x 18. What is the slope formula? Look it up in your textbook. 19. The Pythagorean theorem formula and the distance formula are really the same, however you use the Pythagorean formula, c2 = a2 + b 2 , when you are given 2 d_____________________, and you use the distance formula, (x2 - x1 )2 + ( y2 - y1 )2 , when you are given 2 p____________. Write 2 example problems to show their use. 20. The midpoint of a segment is the point halfway between the e______________ of a segment. x x y y2 sum of x ' s sum of y ' s , ) or ( 1 2 , 1 ) , when you are 21. You use the midpoint formula (pg. 27), ( 2 2 2 2 given 2 p________________, and you are asked to find the midpoint of a s_______________________. Create a problem example and solve it. Find the slope of the line through the given points. 22. A(-3,8), B(4,2) _____ 23. What is the slope of any line parallel to the line through points A and B? _____ 24. What is the slope of any line perpendicular to the line through points A and B? _____ 25. C(1,-3), D(9,-9) _____ 26. What is the slope of any line parallel to the line through points C and D? _____ 27. What is the slope of any line perpendicular to the line through points C and D? _____ 28. E(-2,-3), F(-6,-5) _____ 29. What is the slope of any line parallel to the line through points E and F? _____ 30. What is the slope of any line perpendicular to the line through points E and F? _____ ============================================================================== Chapter 1-1 exercises. Refer to the figure to the right to answer problems 1 - 7. X _______ 1. The line intersecting plane P. ______2. The intersection of AC and XF . A B C ______3. Are points B, F, and X collinear? ______4. Are points A, B, and X coplanar? ______5. Are points A, B, and X contained in Plane P? D F P ____ ____ ____6. Identify 3 non-collinear points ____ ____ ____ ____7. Identify 4 non-coplanar points. j 2 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia Use the midpoint theorem and the segment addition property and the distance formula to solve the following problems. 8. If B is the midpoint of AC and AB = 2x – 3 and BC = 5x – 24, find x, AB, and BC. X = _____, AB = _____, BC = _____ 9. If XB = 14 and XF = 20, find BF. ______ 10. If B is the midpoint of XF and XB = x + 11 and BF = 5x – 1, find x and XF. ____, ____ 11. If AB = 3x, BC = x + 2, and AC = 38, find x and AB. _____, _____ 12. If the coordinate x of G is –8 and the x coordinate of H is 9, find GH. ______ 13. Find the midpoint of the segment having the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ 14. Find the distance between the given endpoints: a. A(-2, -4), B(3, 8) ______ b. C( 3, -4), D( -3, -1) ______ c. E( 2, 1), F(5, 1)_____ d. If the length of PQ is twice the length of AB , then find PQ. _____ e. If the length of RS is one third the length of EF , then find RS. _____ 15. Find the coordinates of A, the missing endpoint, if B(-2, 5) is the midpoint of AC , and the coordinates of C are (-5, 4). See example 5 on page 28. Also do, Pg. 79-80: 2-22 (even); (more practice exercises). ============================================================================== Chapter 1-4, Pg. 36 Angle Measure 1. An angle is formed by two noncollinear rays that have a common endpoint. The rays are called s___________ of the angle. The common endpoint is the v_________________ of the angle and it must always be in the center of the name of the angle. Angles are measure in d_____________. 2. There 3 types of angles: a r____________ angle; it measures ________________ degrees. 3. An a_______________ angle; it measures < 90 degrees, and an o____________ angle; it measures _________ degrees. 4. One could say that there is a fourth type of angle called the straight angle, which is just a line made up of two opposite rays; it measures 180 d__________________. 3 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia 1-5 Angle pair relationships 1. Adjacent angles are 2 angles that have a common v__________ and a common s___________, but no common interior points. Draw an example of 2 adjacent angles and a counterexample. 2. A linear pair is a pair of adjacent angles with noncommon sides that are opposite r_________. Draw an example of a linear pair and a counter example. 3. Vertical angles are two nonadjacent angles formed by two intersecting lines. Draw an example of vertical angles and a counterexample. 4. Complementary angles are two angles, whose m___________________ add up to 180 d________________. Draw an example. 5. Supplementary angles are two angles, whose m___________________ add up to 180 d________________. Draw an example. 6. Perpendicular lines intersect to form f__________ right a________________. Draw a picture that illustrates this. Add the right angle symbol to your drawing. The symbol of perpendicular is ______. ============================================================================= Refer to Figure 2. Matching, you may use more than one letter to describe the angle(s). ________ 1. ________ 2. ________ 3. ________ 4. ________ 5. ________ 6. ________ 7. ________ 8. ________ 9. 1 and 2 1 and 5 3 and 4 1 and BOE 1 and 6 AOF and BOE AOC and COE 2 and 5 4 and AOD a. acute angles b. right angles c. obtuse angles d. adjacent angles e. linear pair f. complementary angles g. supplementary angles h. vertical angles i. congruent angles C D B 3 2 A 4 1 O 6 E 5 F G Figure 2 ============================================================================= 4 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia Refer to figure 2 to solve problems 10 - 17. 10. If m3 = 27, then m4 = _____, and m1 + mBOD = m_____. 12. If m1 = 46 and m4 = 59, then mDOF = _____. 13. If OD bisects COE, then m4 = _____. C D B 3 2 A 4 1 O 14. If OD BF , then m4 + m5 = _____. 15. If OD BF and m4 = 65, then m1 = _____, and m2 = _____, m6 = _____, mAOF = _____. 6 5 F G 16. If OD BF , name all the pairs of complementary angles.______________ _____________________________________________________________ E Figure 2 17. If OD is the bisector of BF , which segments are congruent? __________ 18. Name the vertex of DOF _____________. 19. Write another name for 6 ________________. ============================================================================= Refer to figure 3 to solve problems 18 - 21. 20. Given: m2 = 9x +28 and m3 = 47 – 2x, x = _____, m2 = _____ 2 1 21. Given: m1 = 3x + 5 and m3 = 65, x = _____ 22. Given: m2 = 9x +2 and m4 = 7x + 36, x = _____, m2 = _____ 3 4 Figure 3 23. Given: m1 = x-9 and m2 = 2x, x = _____, m1 = _____ Do problems page 80-81: 24 – 30 (even). 1-6 and 1-7 Two and Three Dimensional Figures 1. A polygon is a closed figure formed by a finite number of c_____________ segments called s___________ such that the sides have a common e_______________ are noncoplanar, and each side intersects exactly 2 other sides, but only at their e_________________. 2. The vertex of each angle is a vertex of the polygon. A polygon is named by the letters of its v____________. Written in the order of the consecutive v________________. 3. A polygon can be c_________________ and convex. 4. A polygon with 4 sides is called a q__________________. One with five sides is called a p_________________. One with n-sides is called an n-gon. In the name Polygon, poly stands for m__________ and gon stands for s________________. 5 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia 5. An equilateral polygon is a polygon in which all s_____________ are congruent, and an equiangular polygon is one in which all a______________________ are c______________________. 6. A convex polygon that is both equiangular and e__________________ is called a r______________ polygon. 7. The perimeter of a polygon is the s________ of the lengths of the s____________. The circumference of a circle is the d_________________ around the circle. 8. The area of a figure is the number of square units needed to cover a s_______________. Review all the formulas on page 58 in your textbook. Draw the figure and write the formula underneath it. 9. Dasan has 32 feet of fencing to fence in a play area for his dog. Which shape of play area uses the modt or all of the fencing and encloses the largest area? a. b. c. d. Circle with radius of about 5 feet Rectangle with length 5 feet and width 10 feet Right triangle with legs of length 10 feet each Square with side length 8 feet 10. Find the perimeter and area of ABC with vertices A(-1, 4), B(-1, -1), and C(6, -1). 11. A rectangle of area 360 sq. meters is 10 times as long as it is wide. Find its length and width. 12. The vertices of a rectangle with side lengths of 10 and 24 units are on a circle of radius 13 units. Find the area between the figures. 13. 6 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia See page 67 to review 3-Dimensional figures. 1. A solid figure with all flat surfaces that enclose a single region of space is called a p________________. Each flat surface or face is a polygon. The line segments where the faces intersect are called e_________. The point where the 3 or more edges intersect is called a v________________. 2. A prism is a polyhedron with two parallel congruent f____________ called b____________ connected by parallelogram faces. Draw one example. 3. A pyramid is a polyhedron that has a polygonal b______________ and 3 or more triangular f_________ that meet at a common v_______________ (peak). Draw one example. 4. A cylinder is a solid with congruent parallel circular b____________ connected by a curved s_____________. Draw a picture. 5. A cone is a solid with a circular base connected by a curved s________________ to a single v____________. Draw a picture 6. A sphere is a set of points in space that are the same distance from a given p____________. A sphere has no faces, no e________, and no v______________. Draw a picture. 7. Find the volume of a cube that has a total surface area of 54 square millimeters. See page 69 for formulas of following 3-D figures: 8. Prism, regular pyramid, cylinder, cone, and sphere. Draw a picture of each figure listed and write the formulas for volume and surface area underneath them. 9. Do problems on page 81-82: 32-43 (all). 7 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia A word problem having to do with the equation of a line. It’s the end of the semester, and the clubs at school are recording their profits. The Science Club started out at $20 and has increased its balance by an average of $10 per week. The Math Club saved $5 a week and started out with $50 at the beginning of the semester. a) Define x and y to fit the problem. b) make a table of values for each club. c) Write an equation for each club. d) Draw a complete graph for each rule and the same axes. e) When do the clubs have the same balance? Show how you can get this number both with the graph and with the equations in c above. f) What is the balance at that point? 8 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia 9 2013-14 Geometry Semester 1 Final Exam Study Guide Chapter 1 FCS, Mr. Garcia ============================================================================================= Chapter 2 Logic & Reasoning Terms: Deductive reasoning, inductive reasoning, conditional, hypothesis, conclusion, converse, contrapositive Conditionals, Converse and Bi-Conditionals A. B. C. D. E. F. G. Restate each of the following given statement into an “if-then” statement. Underline the hypothesis and circle the conclusion. Is the statement true or false? Circle your answer. Write the converse of the conditional and determine whether it is true or false. Write the inverse of the conditional and determine whether it is true or false. Write the contrapositive of the conditional and determine whether it is true or false. If possible, write the bi-conditional statement in “if and only if” form. If not, write a counter example demonstrating why not. 4. Tardy students receive detention. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ 5. All right angles are congruent. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ____________________________________________________________________ 1. A triangle is a polygon that has three sides. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F 10 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 FCS, F. ___________________________________________________________________T or F G. ______________________________________________________________________ 7. Supplementary angles are two angles whose sum is 180. A. & B. ____________________________________________________________ C. T or F D. ___________________________________________________________________T or F E. ___________________________________________________________________T or F F. ___________________________________________________________________T or F G. ______________________________________________________________________ =============================================================== Chapter 3 Topics: parallel Lines & Their Relationships =============================================================== Terms: Parallel (//) lines, transversal, corresponding angles (=), alternate interior angles (=), alternate exterior angles (=), same side (or consecutive) interior angles (sum of 180), (supplementary angles still occur), parallel lines never intersect, parallel lines have the same slope =========================================================================== Refer to figure 4 to determine which lines if any are parallel. 1. Given: 3. Given: 5. Given: 7. Given: 1 5 _____ 2. Given: 8 12 _____ 7 13 _____ 4. Given: 4 14 _____ 6 11 _____ 6. Given: 10 15 _____ 3 and 13 are supplementary _____ Given a b, l m . (Refer to figure 4) b a 3 16 13 5 7 4 1 2 6 14 l 15 11 12 9 8 8. If m 12 = 67 , then m 3 = ______ Figure 4 11 10 m 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 FCS, 9. If m 6 = 108 , then m 16 = ______ 10. If m 4 = 123 , then m 10 = ______ 11. If m 1 = 71 , then m 10 = ______ 12. m1 = 2x + 7 and m16 = x + 30, x = _____, m1 = _____, m16 = _____ 13. m11 = 3x + 6 and m13 = x + 26, x = _____, m11 = _____, m13 = _____ 14. m2 = 11x - 16 and m7 = 7x + 28, x = _____, m2 = _____, m7 = _____ =============================================================== Chapter 4 Topics: Triangle Relationships Term: {classified by angles} right (1 right ), acute (all acute ’s), obtuse(1 obtuse ), equiangular triangles (all 60 angles). {Classified by sides}, Scalene (no sides are =), isosceles (at least 2 sides are =), equilateral triangles (all sides are =) .sum of the interior angles is 180°, sum of the remote interior angles is = to the exterior angle of the triangle, ============================================================== Find the value of x. 1. x = _______ 2. x = _______ 3. x = _______ 70 100 x 70 x x In ABC, find x and mA, then classify the type of triangle according to sides and angels. 4. mA 6 x 24 , mB 2x 7 , and mC x 4 x = ______, mA Classification:____________________ 5. mA = 8x + 9, mB = 3x – 4, mC = 9x + 15 x = ______, mA Classification:____________________ Using the given information, classify each triangle according to its sides and angles. 12 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 FCS, 10. MNO , mM 27 6. DFZ , DF DZ and m D = 90. and mO 82 . 11. LJR , 7. AWV , AW = AV and mA 90. mL 35 and mR 104 . 8. PON , PO = 5, ON = 5, PN = 5. 12. KMN , mM 90, MN = 9. LJI , mL 45 and mI 90 . 13. SYX , mS = 60 and MK. mY = 60. Use the distance formula to classify the triangle by the measure of its sides. 14. A(1, 0) B(3, 3) C(2, 4) AB = _____ BC = _____ AC = _____ Classification ________________ 15. D(4, -6) A(-2, 5) V(0, 7) DA =_____ AV = _____ DV = Classification: _______________ =============================================================== Chapter 4 Topic: Congruent Triangles Term: constructions of congruent triangles, 2 sides and the included angle are (SAS), 2 angles and the included side are (ASA), three congruent sides are (SSS), 2 angles and the non-included side are (AAS), 2 sides and the non-included angle form 2 triangles (SSA). =============================================================== Identify the congruent triangles and justify your answer. If congruency can not be proven write “n p” in both blanks. C F 1. Given: AB ED, BC EF , andCA FD A BAC __________ by _____________. B D E M 2. Given: SM MT , MP MP, and MP bisects SMT MPS __________ by _____________. S P T 13 2013-14 Geometry Mr. Garcia 3. Given: Semester 1 Final Exam Study Guide Chapter 1 OM MN , PR PQ, MO PR, and ON RQ MNO _________ by ______________. FCS, O M Q N P R G F 4. Given: FG JK , GH HK H HJK _________ by _______________. J Given: C is the Midpoint of AD ABC _______by _________________ 5. K A B C E D 6. Given: XZ bisects YXW, YZX is a right angle. XYZ ________ by ________________ Y For the following problems, ABC DEF. 7. Z W X Given: AB = 3y + 12, DE = 5y – 18, find DE. ______ 8. Given: mC = 4y – 23, m F = 2y – 5, find the mC. ______ 9. Use the distance formula to determine whether the triangles with the given vertices are congruent. Given: ∆PQR : P(1,2), Q(3,6), R(6,5) ∆ KLM : K(-2,1), L(-6,3), M(-5,6) PQ = KL = QR = LM = PR = KM = Are they Congruent? Why? Proofs: 14 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 1. Given: a b, l m 2. Prove: mÐ4 @ mÐ10 FCS, m 1 n 2 5 b Statements 14 15 16 1. 9 2 13 a 7 8 4 Reasons 6 10 11 2. 3. 2. Given : AD // BC ; AD BC Prove: ∆ ABD ∆ CDB C D 1 2 Statements Reasons 3 4 B A 1. 2. 3. 4. 5. 9. Use the graph to the right and use the Pythagorean Theorem to determine the length of the longest segment. Round to the nearest hundredth. Be sure to indicate the segment. A D List the segments order from least to greatest. B F C 15 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 FCS, 10. Use the graph to the right to answer the following questions. State the coordinates for an endpoint of the segment with point B as one endpoint and point A as a midpoint. 11. Given: C is the midpoint of BD , BC = (2x – 3)cm and CD = (5x – 24)cm. Find the length of BD . 12. Find the value of x in the figure. 3x + 4 2x + 1 13. If mFBC = 74 and m 1 = 3x - 8 and m 2 = 5x + 26, find x and m 3.(4 points) F A D 2 3 1 B C 14. If m 1 = 41 , and m DOF = 87 , what is m 4? 15. If m 3 = 8x – 12 and m 4 = 4x + 6, and m 1 = 3x – 9, find m 1 6. 16. If 3 4, then OD is a(n) ______________. 17. If GA // BF , their slopes are _______. C B D 2 3 A 1 E 4 O 6 5 F G Figure 2 18. If point B and point D are equidistant from AE , what conclusion can be made about 1 and 4? 19. What is the sum of 1, 2, 3, 4, 5, and 6? =============================================================== 1 y 3 ( x 5) 3 If the equation of line 1 is , state a possible equation which would describe line 2. 16 2013-14 Geometry Mr. Garcia Semester 1 Final Exam Study Guide Chapter 1 FCS, 26. KNG is an isosceles triangle with K as the vertex angle, and KN 5x 2 , and GK 2x 4 . a. Draw a diagram and label the angles and the sides with their lengths in algebraic form. b. What is the length of KN ? …Of KG ? c. For what range of values for GN will the lengths still form a triangle ? d. Make a table of lengths possible for NG . (Use only integers) e. Using the range of values above, find 1 value that will form an ACUTE triangle. Justify using the Pythagorean theorem. f. Using the range of values above, find 1 value that will form an OBTUSE triangle. Justify using the Pythagorean theorem. ============================================================== 27. In QRT, the angles listed from largest to smallest are: a) Q , R , T b) R , Q , T c) T , R , Q d) Q , T , R Q 25 19 R 30 T Why are the following triangles are congruent? Justify your reasoning! Be sure to use the phrase “two sides and the included angle are congruent” instead of SAS! 32. Given : AB CD ; AD BC 33. Given: AE BC ; E C Prove : ∆ABD ∆ CDB D is the midpoint of EC Prove: ∆ADE ∆BDC 34. Determine which postulate can be used to prove the triangles are congruent. If the triangles cannot be proven congruent write NONE. Be sure to write out the postulate (EX: 2 sides and the included angle are congruent instead of SAS) C D A A B B E C D 17