Download Chapter 10: Introducing Geometry

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
10.1.2.4. See worksheet
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. 565-566
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. 567
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. 571-572
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. 575
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. 578
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. 579
10.3.
Problems and Exercises p. 530
10.3.1. Home work: 1, 9-15, 23, 27, 35, 57, 64