Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia 12/8/14

Transcription

1 Name 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 on Thursday of next week. 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 terms. 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?, an angle,, a triangle?, a quadrilateral?, a pentagon?. 2. What does it mean for 3 or more points to be collinear?, Noncollinear?. 3. What does it mean for 3 or more points to be coplanar?, Noncoplanar?. Relationships of lines and planes: 4. What does it mean for 2 lines to be parallel?. 5. What is the symbol for parallel? What is the symbol for perpendicular?. 6. When 2 lines intersect, they intersect at a p. 7. When a line and a plane intersect, they intersect at a p. 8. When 2 planes intersect, they intersect at a l. 9. An angle bisector could be a s, a l, or a r. 10. Any segment, line, or plane that intersects a segment at its m is called a Segment b. 11. 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. 12. 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. 13. When a line segment, a ray, or a line, bisects an angle, it creates two c angles, and their measures are e. 14. What is the difference between an expression and an equation? Write an example of each. 1

2 15. A point where the 2 sides of an angle meet is called? V. Δy 16. The slope of a line can be calculated when you are given 2 p. Δx 17. What is the slope formula? (Look it up in your textbook if you don t remember) 18. The Pythagorean theorem formula and the distance formula are really the same, however you use the Pythagorean formula, c 2 = a 2 + b 2, when you are given 2 d, and you use the distance formula, problems to show their use. (x 2 x 1 ) 2 + ( y 2 y 1 ) 2, when you are given 2 p. Write 2 example 19. The midpoint of a segment is the point halfway between the e of a segment. sum of x ' s sum of y ' s x 20. You use the midpoint formula (pg. 27), (, ) or 1+ x2 y1+ y2 (, ), when you are 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. 21. A(-3,8), (4,2) 22. What is the slope of any line parallel to the line through points A and? 23. What is the slope of any line perpendicular to the line through points A and? 24. C(1,-3), D(9,-9) 25. E(-2,-3), F(-6,-5) 2

3 ============================================================================== 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. 3. Are points, F, and X collinear? 4. Are points A,, and X coplanar? 5. Are points A,, and X contained in Plane P? D A F C P 6. Identify 3 non-collinear points 7. Identify 4 non-coplanar points. j Use the midpoint theorem, the segment addition property, or the distance formula to solve the following problems. 8. If is the midpoint of AC and A = 2x 3 and C = 5x 24, find x, A, and C. X =, A =, C = 9. If X = 14 and XF = 20, find F. 10. If is the midpoint of XF and X = x + 11 and F = 5x 1, find x and XF., 11. If A = 3x, C = x + 2, and AC = 38, find x and A., 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), (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), (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 A, 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 (-2, 5) is the midpoint of AC, and the coordinates of C are (-5, 4). See example 5 on page 28. 3

4 Also do, Pg : 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. 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 counterexample. 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. 4

5 ============================================================================= Refer to Figure 2. Matching, you may use more than one letter to describe the angle(s) and and and and OE 5. 1 and 6 6. AOF and OE 7. AOC and COE 8. 2 and 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 A 1 C O 6 G D 5 F E Figure 2 ============================================================================= Refer to figure 2 to solve problems If m 3 = 27, then m 4 =, and m 1 + m OD = m. 12. If m 1 = 46 and m 4 = 59, then m DOF =. 13. If OD bisects COE, then m 4 =. 14. If OD F, then m 4 + m 5 =. 15. If OD F and m 4 = 65, then m 1 =, and m 2 =, m 6 =, m AOF =. 16. If OD F, name all the pairs of complementary angles. 17. If OD is the bisector of F, which segments are congruent? A 1 C O 6 G D 5 F Figure 2 E 18. Name the vertex of DOF. 19. Write another name for 6. ============================================================================= Refer to figure 3 to solve problems Given: m 2 = 9x +28 and m 3 = 47 2x, x =, m 2 = 21. Given: m 1 = 3x + 5 and m 3 = 65, x = Given: m 2 = 9x +2 and m 4 = 7x + 36, x =, m 2 = Figure Given: m 1 = x-9 and m 2 = 2x, x =, m 1 = 5

6 Do problems page 80-81: (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. 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 most or all of the fencing and encloses the largest area? a. Circle with radius of about 5 feet b. Rectangle with length 5 feet and width 10 feet c. Right triangle with legs of length 10 feet each d. Square with side length 8 feet 10. Find the perimeter and area of AC with vertices A(-1, 4), (-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. 6

7 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. 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. 7

8 9. Do problems on page 81-82: (all). 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

9 ============================================================================================= Chapter 2 Logic & Reasoning Terms: 1. Deductive r uses facts, rules, definitions, theorems, or properties to reach logical conclusions. 2. A counterexample is a f example about a conjecture, it could be a number, a drawing or a statement. 3. I reasoning uses a pattern of examples or observations to make a conjecture, or an intelligent guess. 4. Conjecture is a conclusion reached by using i reasoning. The h is the if part of a conditional statement, and the then part is its c. 5. A conditional statement is a statement that can be written in if-then form. The if part of a conditional statement is called the h, and the then part is called the conclusion. 6. The converse of a conditional is formed by exchanging the hypothesis and c. 7. The inverse is formed by negating both the h and the c. 8. The contrapositive is formed by n both the hypothesis and the conclusion of the converse statement. 9. A postulate or axiom is a statement that is accepted as true without p. 10. A theorem is has to be p for it to be accepted as true. 11. Go to page 127 to read and memorize postulates 2.1 to 2.7. Go to page 144 to read/memorize postulates 2.8 and A 2-column proof is a proof base on deductive r. 13. See page 145 or Theorem 2.2 Properties of Segment Congruence. 14. Page 151 Postulate 2.10 Protractor Postulate Angle Addition Postulate states: m AD + m DC = m AC. Draw a figure that matches the postulate description. Read the following instruction carefully to fill in the blanks below. Each Instruction after the letters corresponds with the letters under each number. A. Restate each of the following given statement into an if-then statement.. Underline the hypothesis and circle the conclusion. C. Is the statement true or false? Circle your answer. D. Write the converse of the conditional and determine whether it is true or false. E. Write the inverse of the conditional and determine whether it is true or false. F. Write the contrapositive of the conditional and determine whether it is true or false. G. If possible, write the bi-conditional statement in if and only if form. If not, write a counter example demonstrating why not. 1. Tardy students receive detention. A. &. C. T or F D. T or F E. T or F 9

10 F. T or F G. 2. All right angles are congruent. A. &. C. T or F D. T or F E. T or F F. T or F G. 3. A triangle is a polygon that has three sides. A. &. C. T or F D. T or F E. T or F F. T or F G. 4. Supplementary angles are two angles whose sum is 180. A. &. C. T or F D. T or F E. T or F F. T or F G. 10

11 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia Chapter 3 Topics: parallel Lines & Their Relationships =============================================================== Terms: If 2 parallel lines are cut by a transversal, then corresponding angles are, alternate i angles are, alternate e angles are, consecutive i angles are s parallel lines never intersect, parallel lines have the same s =========================================================================== Refer to figure 4 to determine which lines if any are parallel. 1. Given: Given: Given: Given: Given: Given: Given: 3 and 13 are supplementary =============================================================== Chapter 4 Topics: Triangle Relationships Term: Triangles may be classified by angles: r triangle, a triangle, o triangle. Equiangular triangles contain all 60 angles, and an eq contains all sides congruent. An i triangle has at least 2 sides e. A scalene has no s equal. All angles of a triangle add up to degrees. The external theorem states that the e angle is equal to the 2 r angles of a triangle. ============================================================== Find the value of x Given a b, l m. (Refer to figure 4) If m 12 = 67, then m 3 = 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. m 1 = 2x + 7 and m 16 = x + 30, x =, m 1 =, m 16 = 13. m 11 = 3x + 6 and m 13 = x + 26, x =, m 11 =, m 13 = 14. m 2 = 11x - 16 and m 7 = 7x + 28, x =, m 2 =, m 7 = a 16 b Figure l 9 10 m 1. x = 2. x = 3. x = x x 11

12 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia In ΔAC, find x and m A, then classify the type of triangle according to sides and angels. x 4. m A = 6x 24, m = 2x 7, and m C = x + 4 x =, m A = Triangle type 5. m A = 8x + 9, m = 3x 4, m C = 9x + 15 x =, m A = Triangle type Using the given information, classify each triangle according to its sides and angles. 6. Δ DFZ, DF < DZ and m D = Δ MNO, m M = 27 and m O = Δ AWV, AW = AV and m A < Δ LJR, m L = 35 and m R = Δ PON, PO = 5, ON = 5, PN = Δ KMN, m M >90, MN = MK. 9. Δ LJI, m L = 45 and m I = Δ SYX, m S = 60 and m Y = 60. Use the distance formula to classify the triangle by the measure of its sides. 14. A(1, 0) (3, 3) C(2, 4) A = C = AC = Classification 15. D(4, -6) A(-2, 5) V(0, 7) DA = AV = DV = Classification: 12

13 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia =============================================================== 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), the hypotenuse and a leg of a right triangle (HL) =============================================================== Identify the congruent triangles and justify your answer. If congruency can not be proven write n p in both blanks. C F 1. Given: A ED, C EF, andca FD ΔAC Δ by. A D E 2. Given: SM MT, MP MP, andmp bisects SMT M ΔMPS Δ by. S P T 3. Given: OM MN, PR PQ, MO PR, and ON RQ O Q M ΔMNO Δ by. 4. Given: FG JK, GH HK ΔHJK Δ by. P F J G H K N R 5. Given: C is the Midpoint of ΔAC Δ by A C 6. Given: bisects YXW, YZX is a right angle. ΔXYZ Δ by D Y E Z W X 13

14 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia For the following problems, ΔAC ΔDEF. 7. Given: A = 3y + 12, DE = 5y 18, find DE. 8. Given: m C = 4y 23, m F = 2y 5, find the m C. 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? ============================================================== 10. Proofs: 1. Given: a b, n m 2. Prove: 4 10 Statements Reasons b 4 1 m 2 n a a

15 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia 11. Given : AD C ; AD C 1. Prove: AD CD Statements Reasons D 2 A 1 4 C Use the graph to the right and use the a) Pythagorean Theorem to determine the length of the longest segment. Round to the nearest hundredth. e sure to indicate the segment. b) List the segments order from least to greatest. D C A F 14. Use the graph to the right to answer the following questions. State the coordinates for an endpoint of the segment with point as one endpoint and point A as a midpoint. 15. Given: C is the midpoint of D, C = (2x 3)cm and CD = (5x 24)cm. Find the length of D. 16. Find the value of x in the figure. 3x + 4 2x

16 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia 17. If m FC = 74 and m 1 = 3x - 8 and m 2 = 5x + 26, find x and m 3.(4 points) A F C D 18. If m 1 = 41, and m DOF = 87, what is m 4? 19. If m 3 = 8x 12 and m 4 = 4x + 6, and m 1 = 3x 9, find m If 3 4, then OD is a(n). 21. If GA // F, their slopes are. 22. If point and point D are equidistant from AE, what conclusion can be made about 1 and 4? A 1 C 2 3 O 4 6 G 5 D Figure 2 F E 23. What is the sum of 1, 2, 3, 4, 5, and 6? =============================================================== 24. Δ 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 OTUSE triangle. Justify using the Pythagorean theorem. ============================================================== 25. In ΔQRT, the angles listed from largest to smallest are: a) Q, R, T b) R, Q, T c) T, R, Q d) Q, T, R T Q R 16

17 Geometry Semester 1 Final Exam Study Guide FCS, Mr. Garcia Why are the following triangles are congruent? Justify your reasoning! e sure to use the phrase two sides and the included angle are congruent instead of SAS! 26. Given : A CD; AD C 27. Given: AE C ; E C Prove : AD CD D is the midpoint of EC Prove: ADE DC 28. Determine which postulate can be used to prove the triangles are congruent. If the triangles cannot be proven congruent write NONE. e sure to write out the postulate (EX: 2 sides and the included angle are congruent instead of SAS) C D A A E D C 17