Chapter 10: Introducing Geometry 10.1 Basic Ideas of Geometry Geometry is everywhere o Road signs o Carpentry o Architecture o Interior design o Advertising o Art o Science Understanding and appreciating geometry is basic to understanding and appreciating mathematics Focus for this chapter is on visualizing o Two-dimensional figures and their properties o Three-dimensional figures and their properties o Further development of your spatial sense 10.1. Seeing Geometry in the World 10.1.1. Geometry in nature 10.1.1.1. honeycombs 10.1.1.2. snowflakes 10.1.2. Fibonacci sequence 10.1.2.1. 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 10.1.2.2. sunflowers 10.1.2.2.1. Ratio of counterclockwise spirals to clockwise spirals is often 55:34 or 34:21
10.1.2.3. pine cone Ratio = 13:8 or 8:5 10.1.3. Golden ratio 10.1.3.1. Approximately 1.618 10.1.3.2. Ratio of successive Fibonacci numbers 10.1.3.3. star fish 10.1.3.4. snail shell 10.1.4. Geometry in human endeavors 10.1.4.1. geometry = earth measure 10.1.4.2. Egyptian pyramids 10.1.4.3. Pentagon 10.2. Modeling and Defining Basic Geometric Ideas 10.2.1. Points, lines, planes, and space 10.2.1.1. point no length, no width, no height 10.2.1.2. space the set of all points that has no boundaries 10.2.1.3. line points in a straight, unlimited length with no width, height, or endpoints 10.2.1.3.1. Two different points define exactly one line 10.2.1.4. plane set of all points on a flat surface with no height and no edges
10.2.1.4.1. Three points not contained in exactly one line define exactly one plane 10.2.1.5. See figures p. 507-508 10.2.1.6. collinear points contained on one line 10.2.1.7. coplanar points contained in one plane 10.2.1.8. intersect 10.2.1.8.1. lines two distinct lines intersect in exactly one point called the point of intersection 10.2.1.8.2. planes two distinct planes intersect in exactly one line called the line of intersection 10.2.1.9. parallel two distinct lines in the same plane that do not intersect 10.2.1.10. skew two distinct lines in two different planes that do not intersect 10.2.2. Segments, rays, angles 10.2.2.1. See table 10.1 p. 517 10.2.2.2. line segment set of points A and B ad all of the points between A and B 10.2.2.3. ray point A and all of the points on AB on the same side of A as point B 10.2.2.4. angle the union of two rays with a common endpoint 10.2.2.5. length measure assignment of a real number of some unit to a segment 10.2.2.6. congruent line segments have the same length 10.2.2.7. midpoint of a segment M is the midpoint of EF if and only if EM MF 10.2.2.8. bisector of the segment any point, line segment, ray, line or plane that contains the midpoint of the segment 10.2.2.9. degree measure real number between 0 and 360 degrees that defines the amount of rotation or size of an angle 10.2.2.10. protractor a device for measuring angles 10.2.2.11. straight angle 180 angle 10.2.2.12. reflex angle > 180, but < 360 10.2.2.13. zero angle - 0 or no rotation 10.2.2.14. interior points inside angle
10.2.2.15. exterior points outside angle 10.2.2.16. angle bisector an interior ray that divides the measure of the angle into two congruent angles 10.2.3. Special angles and perpendicular lines 10.2.3.1. See table 10.2 p. 521-522 10.2.3.2. right angle 90 10.2.3.3. acute angle 0 < angle < 90 10.2.3.4. obtuse angle 90 < angle < 180 10.2.3.5. complementary angles sum of two angles is 90 10.2.3.6. supplementary angles sum of two angles is 180 10.2.3.7. adjacent two angles with same vertex and a common side, with no common interior points 10.2.3.8. linear pair pair of adjacent angles with two non-common sides on the same line, also form supplementary angles 10.2.3.9. vertical angles pair of angles formed by two intersecting lines and that are not a linear pair of angles 10.2.3.10. perpendicular two lines are perpendicular if and only if they intersect to form four right angles 10.2.3.11. perpendicular bisector of a segment is a perpendicular line which passes through the midpoint of the line segment 10.2.4. Circles and polygons 10.2.4.1. open curve path in a plane with different starting and ending points 10.2.4.2. closed curve path in a plane with the same starting and ending point 10.2.4.3. non-simple closed curve closed curve that has more points in common than the starting and ending points 10.2.4.4. simple closed curve closed curve that only has the beginning and ending points in common 10.2.4.5. circle special simple closed curve where all points in the curve are equidistant from a given point in the same plane 10.2.4.6. center middle point of the circle 10.2.4.7. chord line segment connecting two distinct points on the circle 10.2.4.8. diameter is a chord that passes through the center of the circle 10.2.4.9. radius line segment connecting the center of the circle to any point on the circle 10.2.4.10. polygon closed curve created by the union of line segments meeting at their endpoints such that 10.2.4.10.1. at most, two segments meet at one point 10.2.4.10.2. each segment meets exactly two other segments at their endpoints 10.2.4.11. non-simple polygon at least one pair of line segments intersect in a point other than their endpoints 10.2.4.12. simple polygon follows polygon rules 10.2.4.13. simple convex polygon line test: draw a line through the polygon; No line can be drawn such that the line is intersected by the polygon more than twice
10.2.4.14. simple non-convex (concave) polygon - line test: draw a line through the polygon; At least one line can be drawn such that the line is intersected by the polygon more than twice 10.2.4.15. n-gon the whole number n represents the number of sides for the polygon: a triangle is a 3-gon; a square is a 4-gon 10.2.4.16. interior angle formed by two sides of the polygon with a common vertex 10.2.4.17. regular polygon simple polygon where the all the line segments and all of the angles are congruent 10.2.4.18. See table 10.3 p. 525 10.2.5. Triangles 10.2.5.1. Union of three line segments formed by three distinct non-collinear points 10.2.5.2. vertices intersection points of line segments forming the angles of the polygon 10.2.5.3. sides the line segments forming the polygon 10.2.5.4. median line segment connecting a vertex to the midpoint of the side opposite the vertex 10.2.5.5. altitude line segment from a vertex of a triangle to a line containing the side of the triangle opposite the vertex 10.2.5.6. See table 10.4 p. 527 10.2.6. Quadrilaterals 10.2.6.1. Square and a rectangle are special types of kites? 10.2.6.2. See table 10.5 p. 528 10.3. Problems and Exercises p. 530 10.3.1. Home work: 1, 9-15, 23, 27, 35, 57, 64