Points, Lines, and Planes

Even though there is no formal definition for these terms, there is general agreement of their meaning: A point is a dimensionless location in space. It is represented by a dot and named with an uppercased letter. A line has one dimension. It extends infinitely in both directions and is represented by a line with arrowheads, as shown. The following are all correct names for the line shown: The above line can be called line m, Line AB, Line BC, Line CA, Line CB, etc. In symbols this is: AB, CB, AC, etc. A plane has two dimensions. It is represented by a parallelogram or a trapezoid to show perspective. It is named by 3 points on the plane, or by a single uppercased letter. The plane to the right can be named in several different ways. Example: Plane DEF or Plane FDE m A B C D N E F

Collinear points are points that are on the same line. Coplanar points are points that are on the same plane. A ray is the union of a half-line and the point that created it. That point is said to be the endpoint of the ray. A line segment (or just a segment) is the intersection of two rays with distinct endpoints, each of whose endpoints are included in the other’s half-line. Two rays are opposite rays if their union is a line and their intersection consists of exactly one point. 1. Do MA and AM refer to the same line segment? 2. Do AT and TA refer to the same ray? M A T H

Questions Give two other names for BD. 4. Give another name for plane n. 5. Name three points that are collinear. 6. Name four points that are coplanar. E. 7. Name all rays with endpoint Q. Which of these rays are opposite rays? p T S R Q h.A.A B.C.C.D.D E..F.F G.F.F

In Geometry, a rule that is accepted without proof is called a postulate or axiom. A rule that can be proved is called a theorem. 1. Distance Postulate: To any two distinct points, there exists a real unique real number called the distance. a. Distance is always non-negative. b. The distance from point A to point B is denoted by AB. The length, or measure of a line segment, AB, for example, is also AB. c. Note that we never discuss the length of a line because its length is infinite. d. If AB = 0, then A and B must be the same point. This is known as a zero segment. 2. Ruler Postulate: The points on a line can be put in a one-to-one correspondence with the real numbers. The real number corresponding to a point is called its coordinate. The distance between points A and B, written AB, is the absolute value of the difference of the coordinates of A and B. Every point has a number Every number has a point This implies a line is infinitely dense, that is there is no way to pass from one half plane created by a given line to the other half plane created by that line without passing through a point of the line a. Ex: If the coordinate of A is a and the coordinate of B is b, then AB = |a – b| or |b – a| b. Ex: If the coordinate of A is a and the coordinate of B is 7, then AB = |a – 7| c. Ex: If the coordinate of A is a and the coordinate of B is 7, and AB = 10, find the value(s) of a:

ACB ABBC AC If A, B, and C are collinear in that order, then B is between A and C. Segment Addition Postulate: If B is between A and C, then… a. AC = AB + BC b. AB = AC – BC c. BC = AC – AB Segments are congruent if they have the same length: AB = CD iff a.When writing up proofs later in the course, we will often go from “ = ” to “ ” or vice/versa b. It is incorrect to write that or that ; Equals is for numbers, congruence is for shapes.

8. Use the Segment Addition Postulate to find XZ. Questions 8-9 2x+7 3x-7 11x Use the Segment Addition Postulate to find XZ.

Find the distance between A(4,8) and B(1,12) A (4, 8)B (1, 12) Using the DISTANCE FORMULA to find the distance between two points.

Find the midpoint between A(4,8) and B(1,12) A (4, 8)B (1, 12) Using the MIDPOINT FORMULA to find the midpoint of two points.

10. Find the distance between: (-5, 8) and (2, - 4) 11. Find the midpoint between: A) (-5, 8) and (2, - 4) Questions 10-11

Angles

Measure and Classify Angles Describe Angle Pair Relationships Objectives: To define, classify, draw, name, and measure various angles To use the Angle Addition Postulates To use special angle relationships to find angle measures

Angle angle sides vertex  An angle consists of two different rays (sides) that share a common endpoint (vertex). Sides Vertex This angle can be called: Angle ABC, <ABC, <CBA OR it can be named by the vertex like so: <B

Types of Angles Go to the following website to learn about the different types of angles. Don’t forget to press the play button on top of the picture to see the animation.

Classifying Angles Questions Use the picture to the right to identify each of the following. 13.An acute angle: 14.An obtuse angle: 15.A right angle: 16.A straight angle: A B C D E F G

Questions Use the diagram to find the measure of the indicated angle. Then classify the angle. 17.  KHJ 18.  GHK 19.  GHJ 20.  GHL

21. What is the measure of  DOZ?... How did you get to your answer?

Angle Addition Postulate If P is in the interior of  RST, then m  RST = m  RSP + m  PST.

22. Given that m  LKN = 146°, find m  LKM and m  MKN.

Congruent Angles congruent angles  Two angles are congruent angles if they have the same measure.

Angle Bisector angle bisector An angle bisector is a ray that divides an angle into two congruent angles.

23. A B C D Ray BD bisects <ABC. Solve for x.

24. In a diagram, YW bisects  XYZ. m  XYW = ° and m  ZYW= (-18m)°. Find m  XYW. Hint: Draw the picture Hint: Angle may not be drawn to scale

C Comes Before S… Complementary Angles Sum to 90 Degrees Supplementary Angles Sum to 180

Linear Pairs of Angles

linear pair  Two adjacent angles form a linear pair if their noncommon sides are opposite rays. supplementary  The angles in a linear pair are supplementary.

Vertical Angles

vertical angles  Two nonadjacent angles are vertical angles if their sides form two pairs of opposite rays.  Vertical angles are formed by two intersecting lines.

25. Solve for x.