Geometry Definitions

1. Geometry Definitions Marie Bruley Math E-Geometry

2. Points • Three Points C, M, Q • 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).

3. Lines • A line(straightline) It extends forever in two opposite directions. A line has infinite length, zero width, and zero height. The symbol ↔ written on top of two letters is used to name a line. A line may also be named by one small letter l.

4. Types of Lines • PARALLEL lines- two lines that are always the same distance apart, and will never intersect. Parallel can be abbreviated as ||. An example of parallel lines is on the Italian flag. Lines a and b on the flag are parallel.

5. Types of Lines • PERPENDICULAR LINES – two lines that intersect and form angles measuring exactly 90 degrees, like the edges of a building. If an angle measures 90 degrees, a square is place where the lines intersect to show that it is a right angle. Perpendicular is often abbreviated as _|_. • Line a _|_ b reads as line a is perpendicular to line b.

6. Plane • A plane may be considered as a set of points forming a flat surface.

7. Line Segment • We may think of a line segment as a "straight" line that we might draw with a ruler on a piece of paper. A line segment does not extend forever, but has two distinct endpoints. We write the name of a line segment with endpoints A and B as . Note how there are no arrow heads on the line over AB such as when we denote a line or a ray. B A

8. Intersecting lines • Intersecting lines come together at a point. Example point M.

9. Ray • We may think of a ray as a "straight" line that begins at a certain point and extends forever in one direction. • The point where the ray begins is known as its endpoint. • We write the name of a ray with endpoint A and passing through a point B as .

10. Angles • Two rays that share the same endpoint form an angle. The point where the rays intersect is called the vertex of the angle. The two rays are called the sides of the angle.

11. Angles, Cont. • We can name an angle by using a point on each ray and the vertex. The angle below may be named angle ABC or as angle CBA; you may also see this written as or as

12. Degrees: Measuring Angles • We measure the size of an angle using degrees. • The Protractor Postulate

13. Types of Angles • Acute Angle: Measures between 0° and 90° • Right Angle: Measure of 90° • Obtuse Angle: Measure between 90° and 180° • Straight Angle: Measure of 180°

14. Angle Relationships • Complementary Angles: Two angles are called complementary angles if the sum of their degree measurements equals 90 degrees. • Supplementary Angles: Two angles are called supplementary angles if the sum of their degree measurements equals 180 degrees.

15. Angle Relationships • Congruent Angles: Angles with equal measures. • Adjacent Angles: Share a vertex and a common side but no interior points. • Bisector of an angle: a ray that divides the angle into two congruent angles. In this picture is the angle bisector.

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17. Postulates • A statement that is accepted without proof. • Usually these have been observed to be true but cannot be proven using a logic argument.

18. Postulates Relating Points, Lines, and Planes • Postulate 5: A line contains at least two points; a plane contains at least three points not all in one line; space contains at least four points not all in one plane.

19. Postulates Relating Points, Lines, and Planes • Postulate 6: Through any two points there is exactly one line.

20. Postulates Relating Points, Lines, and Planes • Postulate 7: Through any three points there is at least one plane (if collinear), and through any three non-collinear points there is exactly one plane.

21. Postulates Relating Points, Lines, and Planes • Postulate 8: If two points are in a plane, then the line that contains the points is in that plane. B . A .

22. Postulates Relating Points, Lines, and Planes • Postulate 9: If two planes intersect, then their intersection is a line.

23. Theorems • Theorems are statements that have been proven using a logic argument. • Many theorems follow directly from the postulates.

24. Theorems Relating Points, Lines, and Planes • Theorem 1-1: If two lines intersect, then they intersect in exactly one point. • Theorem 1-2: Through a line and a point not in the line there is exactly one plane. • Theorem 1-3: If two lines intersect, the exactly one plane contains the lines.