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# Geometry

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1. Geometry

2. Points, Lines, and Planes • A point is the most fundamental object in geometry. It is represented by a dot and named by a capital letter. A point represents position only; it has zero size (that is, zero length, zero width, and zero height). B

3. A line (straight line) can be thought of as a connected set of infinitely may points. It extends infinitely far in two opposite directions. A line has infinite length, zero width, and zero height. Any two points on the line name it. KL K L N

4. Points that lie on the same line are called collinear points. If there is no line on which all of the points lie, then they are noncollinear points. B C A Y X Z

5. A plane may be considered as an infinite set of points forming a connected flat surface extending infinitely far in all directions. A plane has infinite length, infinite width, and zero height (or thickness). It is usually represented in drawings by a four-sided figure. A single capital letter is used to denote a plane. The word plane is written with the letter so as not to be confused with a point.

6. R Plane R T Plane T

7. Segments, Midpoints, and Rays • A line segment is a connected piece of a line. It has two endpoints and is named by its endpoints. Sometimes, the symbol – written on top of two letters is used to denote the segment. A B C D BC

8. A midpoint of a line segment is the halfway point, or the point equidistant from the endpoint. • A ray is also a piece of a line, except that it has only one endpoint and continues forever in one direction. It could be thought of as a half-line with an endpoint. AB A B OP O P

9. Angles • Two rays that have the same endpoint form an angle. That endpoint is called the vertex, and the rays are called the sides of the angle. In geometry, an angle is measured in degrees from 0 to 180. The number of degrees indicates the size of the angle. ABC A C B

10. Different names for the same angle • Angle 3 Example IMJ or JMI I J H 3 K 2 4 1 5 G L M

11. Angle Bisector • An angle bisector is a ray that divides an angle into two equal angles. X Y O Z OY is bisector of XOZ because = m XOY = m YOZ

12. Right Angle • A right angle has a measure of 90. The symbol in the interior of an angle designates the fact that a right angle is formed. 90

13. Acute Angle • An acute angle is any angle whose measure is less than 90. Less than 90 degrees

14. Obtuse Angle • An obtuse angle is an angle whose measure is more than 90. but less than 180. More than 90 degrees and less than 180 degrees

15. Straight Angle • Some geometry texts refer to an angle with a measure of 180 as a straight angle. 180 Degrees

16. Recognizing Different Angles • CFE • CFD • AFE • AFD B C D 130 90 40 180 A F E

17. Adjacent Angles • Adjacent angles are any two angles that share a common side separating the two angles and that share a common vertex. Below angle 1 and 2 are adjacent angles. A 1 D 2 B C

18. Vertical Angles • Vertical angles are formed when two lines intersect and form four angles. Any two of these angles that are not adjacent angles are called vertical angles. Line l and line m intersect and point Q, forming angle 1, angle 2, angle 3, and angle 4. l Q 1 2 4 3 m

19. Vertical Angles • Angle 1 and Angle 3 • Angle 2 and Angle 4 • Adjacent Angles • Angle 1 and Angle 2 • Angle 2 and Angle 3 • Angle 3 and Angle 4 • Angle 4 and Angle 1

20. Complementary Angles • Complementary angles are any two angles whose sum is 90 degrees. 45 45 + 45 = 90 45

21. Supplementary Angles • Supplementary angles are two angles whose sum is 180. 135 45 135 + 45 = 180

22. Intersecting Lines • Two or more lines that meet at a point are called intersecting lines. That point would be on each of these lines. Below lines l and m intersect at Q l Q m

23. Perpendicular Lines • Two lines that intersect and form right angles are called perpendicular lines. The symbol is used to denote perpendicular lines. Below line l line m. l m

24. Parallel Lines • Two lines, both in the same plane, that never intersect are called parallel lines. Parallel lines remain the same distance apart at all times. The symbol is used to denote parallel lines. Below l m. l m

25. Parallel Planes • Parallel planes are two planes that do not intersect.

26. Perpendicular Planes • A line l is perpendicular to plane A is l is perpendicular to all of the lines in plane A that intersect l. (Think of a stick standing straight up on a level surface. The stick is perpendicular to all of the lines drawn on the table that pass through the point where the stick is standing). • A plane B is perpendicular to a plane A if plane B contains a line that is perpendicular to plane A. (Think of a book balanced upright on a level surface.)

27. Perpendicular Planes C Plane T A B Plane S D