POINTS, LINES, PLANES, AND ANGLES, Study notes of Geometry

Download POINTS, LINES, PLANES, AND ANGLES and more Study notes Geometry in PDF only on Docsity! Wilkinson Page 1 of 30 Geometry POINTS, LINES, PLANES, AND ANGLES Chapter 1 ________________________ NAME Wilkinson Page 2 of 30 1-2 Points, Lines, and Planes 3 Undefined terms in Geometry: 1. 2. 3. POINT A Wilkinson Page 5 of 30 SPACE – INTERSECTION - The intersection of 2 lines is a _____________. The intersection of 2 planes is a ___________. The intersection of a plane and a line not on that plane is a _____________. 1-2 Points, Lines, and Planes (continued) intersection _ the set of points in both figures (Dashes in the diagrams indicate parts hidden from view.) Z ison AB. AB contains Z. <> AB passes through Z. ee AB and CD intersect at Z. << Plane M contains AB and Y. CD intersects Mat Z. M and N intersect in EF, EF is the intersection of M and N. — M and N contain EF. Classify each statement as true or false. 1. BCis in plane M. < 2, Plane M contains AB. 3. Line / intersects 4B at point B. 4. AB and DA intersect at.A. 5. ADis in plane M. <> 6. Plane M intersects AE at point B. 7. AE intersects plane M at point B. 8. A, B, and E are collinear. 9. B, F, and D are collinear. 10. A, B, and C are coplanar. 11. B, C, F, and G are coplanar. 12. A, B, C, and G are coplanar. 13. A, B, C, and F are coplanar. The plane that contains the shaded region can be called plane ABCD. . « . ree G 14. Name three lines that intersect at point IG. i 15, Name two planes whose intersection is FR. EB ‘p | 16. Name the intersection of plane EHGF and plane EFBA. 17, Name two planes that do not intersect. 18, Are points D, H, G, and C coplanar? 19. Are points D, H, G, and F coplanar? 20. Are points 4, B, G, and HA coplanar? Sketch and label the figures described. Use dashes for parts hidden from view. 21, Line.AB intersects plane X at point C. 22. Two planes M and N intersect in line /. 23. Horizontal plane P contains two lines RS and TU that intersect at point O. Wilkinson Page 6 of 30 Points, Lines, and Planes For use after Section 1-2 Estimate to compare the values, 1, The distance from U to S and the distance 2. The area of figure I (area 1) and the area from UtoT of figure (area) Tr 7 7 6 U I Classify each statement as true or false. (Write 7 or F) > —- 3. AB isin plane MM, _ 4, M contains CD. <> _— — S$. AB intersects Nat EB _ 6. AB intersects CD at D, — 7. Fis in plane N. 8. BisinplaneN 9. A, D, and F are coplanar. 10. N contains G D, & and F —., _ 4 it. Bison CD. __ 12. C, D, and E are collinear, Exs, 3-12 Name a fourth point that is in the same plane as the given points. A E 3.4,RF 4.6 WG 2B mF 15.GGE___. 6. BC pk-f----.----- i No = G Name each of the following. c Fxs. 13-20 17. Three lines that intersect at point E 18. The plane that does not intersect plane FGHE 19. Two planes that intersect in eG —_ 20. Three planes that intersect at point D Sketch and label the figures described. 24. Vertical planes P and Q intersect in RS. 22. Horizontal plane M containing AB > intersects AC at point A. Wilkinson Page 7 of 30 Wilkinson Page 10 of 30 Wilkinson Segment Addition Postulate: If B is between A & C, then AB + BC = _____. A B C Wilkinson Example 1 AC = _____ A B C 4 8 Wilkinson Page 11 of 30 Wilkinson Example 2 If AC = 16, find x. A B C 2x X + 7 Wilkinson Example 3 If AC = 30, find BC. A B C 3x 4x + 2 Wilkinson Page 12 of 30 Wilkinson Congruent – objects that have the same ________ and __________  Congruent Segments – segments that have __________ ____________ If DE = FG, then .FGDE  Wilkinson Midpoint of a Segment: the point that ______________ the segment into ____ ____ segments X Y Z 2 2 1-3 Segments, Rays, and Distance Objectives: Use symbols for lines, segments, rays, and distances; find distances. Use the Segment Addition Postulate. In learning a new language, the first things you need to learn are vocabulary and rules of grammar. In geometry, you need vocabulary, symbols, and rules called postulates. segment A segment is named by gi giving its its endpoints. Xand Z are x Zz the endpoints of XZ. XZ and ZX are th the same segment. ¥ is between X and Z. ¥ must be on XZ. x Y Z +e _____»__e» ray A ray is named by giving its endpoint and another point on the ray. The endpoint of a ray is always named first. xX y Zz XY and XZ are the same ray. — = . XZand ZX are different rays. —5 —> . x y Zz YX and YZ are opposite rays. oo Refer to the diagram at the right. 4 1. Give several names for the line. 2. Name several segments in the figure. << 3. Name several rays in the figure. 4, Name two pairs of opposite rays. Classify each statement as true or false. 5. Cis between A and B. 6. AB and AGare oppesite rays. 7. CBis the same as BC. 8. CBis the same as BC. 9. CBi is the same as BC. 10. Bi is the same as 11. JFis the same as CB. 12. BAis the same as BD. length XZ is the length of XZ or the distance between point X and point Z. You can find the length of a segment on the number line by computing the absolute value of the difference of the coordinates of the endpoints. Length must be a positive number. Example 1 y Xx Zz Find XZ and YX. + -4-3-2-10 1234567 Solution XZ = |2-5| = |-3| =3 YX = |-3-2| = |-5| =5 or XZ = |5—2| = |3] =3 or YX = |2-(-3)| = [5] =5 Segment Addition Postulate If B is between 4 and C, then AB + BC = AC. Wilkinson Page 15 of 30 1-3 Segments, Rays, and Distance (continued) SR + RT = ST (by the Segment Addition Postulate) x+x+4=14 Example 2 Ris between S and T, with RT = x, ! SR =x + 4,and ST = 14, es, Find the value of x. Then find RT and SR. x+ 48 Solution 2x+4= 14 SR=x+4 2x = 10 =5+4 x=5 RT=x=5 =9 congraent Two objects that have the same size and congruent segments have the same length. TfAB = CD, then AB = CD. i A Cc shape are congruent (=). For example, ——! | - 5 D midpoint of a segment A midpoint divides a segment into two A congruent segments. —T7T X is the midpoint of AB, so AX = XB. bisector of a segment A segment bisector is a linc, segment, ray, or plane that intersects a segment at its midpoint. AX = XB, so plane M, BS, and VW are all bisectors of AB. For Exercises 13-16, refer to the number line at the right. 13. Find BD. A OB C DE 14, Find the length of AC. rt Trt 15. Find the distance between B and E. - “Eo ras . 16. Find the coordinate of the midpoint of AE. 17. Find the value of x. 18. In the diagram, AC = CE and B is the midpoint of AC. CD = 2and AB = 3. Find BC, AC, and DE. | 183 A " eo ot. 4 B CD E x B A Wilkinson Page 16 of 30  CARD CARD CARD 1.4 ANGLES A Angle - figure formed by 2 rays that have the same endpoint The rays are the The common endpoint is the 4 . Cc When naming an angle, use letters, letter, or number. 3 Name the angle. Vertex = Sides = and Name 3 angles. A D B c Angles are measured in There are 4 classifications of angles. CARD CARD Wilkinson Page 17 of 30  CARD bisector of an angle -a that divides an angle into 2 angles Ex. Given: EC bisects “BED, mZAEB = 19x, mZBEC = 8x + 20 Find: x &mZCED A Day 2: P. 21 #2, 15-18, 29-34 (draw picture for each) Wilkinson Page 20 of 30 Wilkinson Page 21 of 30 Wilkinson Page 22 of 30 Wilkinson Page 25 of 30 Wilkinson Page 26 of 30 Practice 3 | Lessons 4-3 through 1-5 Definitions and Postulates Refer to the diagram and name each of the following. 1. An angle adjacent to POT 2. The ray opposite to TS 3. An obtuse angle 4. The sides of ZTOR and Pp QA 5. Two right angles and 6. A point on PG that is not on PO 7. The vertex of the 20° angle 8. The point between P and R Classify each statement as true or false. 9. Through any two points there is exactly one line. ‘ _____ 10. Through any three points there is exactly one line. 11. Through any three points there is exactly one plane. 12. Two lines intersect in exactly one point. 13. Two planes intersect in exactly one point. _ 14, Two planes intersect in a line. 15. A line and a plane can intersect in a point. Complete each statement with the word always, sometimes, or never. 16. Adjacent angles are congruent. 17. If points A and & are in plane R and point C is on AB, then C is ink. 18. Two intersecting lines lie in exactly one plane. 19, A line and a point not on the line lie in more than one plane. 20. Aline _ contains at least two points. Wilkinson Page 27 of 30