Thursday, January 31, 2008

Geometry Vocabulary for Proofs

Vocabulary - Euclidean Geometry:

1. Space - is a set of points that forms a completely flat surface extending indefinitely in all directions.

2. Collinear Points - is a set of points all of which lie on the same straight line.

3. Coplanar points - is a set of points all of which lie on the same plane.

4. Betweenness of Points or Segment/Angle Addition Postulate (we will call this Partition Postulate) - point B is between point A and point C, if A, B and C are distinct collinear points and AB + BC = AC. We shall say that the part plus the part equals the whole (part + part = whole)
5. Segment or line segment - is a set of points consisting of two points on a line, called endpoints, and all the points on the line between the endpoints.

6. Length of a line segment - is the distance between the endpoints.

7. Congruent segments - are segments that have the same measure.
Congruent angles - are angles that have the same measure.

8. Midpoint of a segment - is the point of the line segment that divides the segment into two congruent segments or divides the line segment in half.

9. Bisector of a Segment or angle - is any line, ray, or point that intersects the segment or angle at its midpoint.

10. Rays - is a part of a line that consists of a point on the line called the endpoint and all the points on one side of the endpoint
11. Opposite Rays - are two rays of the same line with a common endpoint and no other point in common.

12. Angle - is a set of points that is the union of two rays having the same endpoint or vertex.
13. Sides of an angle - are the rays that make up the angle

14. Vertex of an angle - is the common endpoint of the two sides of an angle.
15. Adjacent angles - are two angle in the same plane that have a common vertex and a common side but do not have any common interior points.

16. Exterior sides of adjacent angles - the two sides of adjacent angles that are not common to both angles.

17. Vertical angles - are two angles in which the sides of one angle are opposite rays to the sides of the second angle. Vertical angles are congruent.

18. Addition Postulate - if A = B, then A + C = B + C

19. Subtraction Postulate - if A = B, then A - C = B - C

20. Multiplication Postulate - if A = B, then AC = BC

21. Division Postulate - if A = B and C does not equal zero, then A/C = B/C

22. Halves Postulate - if A = B, then A/2 = B/2

23. Doubles Postulate - if A = B, then 2A = 2B

24. Substitution Postulate - if A = 5 and A + B = 12, then 5 + B = 12.
- you plug in the equal value
25. Reflexive Postulate - A = A

26. Symmetric Postulate - if A = B, then B = A.

27. Transitivity Postulate - if A = B AND B = C, then A = C
OR if A is less than B AND B is less than C, then A is less than C.

28. Distributive Postulate - if A(B + C) then AB + AC

Other Vocabulary:

Congruent Angles - angles that have the same measure.

Complementary Angles - are two angles that the sum of their measures is 90 degrees

Supplementary Angles - are two angles that the sum of their measures is 180 degrees.

Perpendicular Lines - form right angles.

Right angle - has a measure of 90 degrees.

Linear pair - are two adjacent angles where their noncommon side are opposite rays.

Here is one way how to write a proof:

1. You set up a t-table and write the word Statements in the left column and Reasons in the right column.

2. You are given a statement with another statement to prove based on the given statement.

3. You write the given statement in the left column with the word given for your reason. Then you write statements that support what you are given with logically reason using vocabulary, postulates or already proven theorems.

Here are a few examples of algebraic proofs:

As you can see, you have been doing proofs, just not formally for awhile. We will continue these notes as we go along in our learning of proof writing. This is for a basic class in High School. There are many more ways to write proofs along with many more vocabulary words.