1.3 Collinear, Betweenness, and Assumptions Date: Goals: How are diagrams interpreted in the study of geometry

Collinear/Noncollinear: Three or more points that lie on the same line are collinear.

Points that do not lie on the same line are noncollinear.

Two points are always collinear.

For a point to be between two other points, all three points must be collinear.

Example 1: Use the diagram to the right to answer the following questions. a. Are points A, B, and C collinear? A Yes, B is between A and C. b. Is point B between A and D? B No, the three points are noncollinear. c. Name 3 noncollinear points. AEC, , and E D C d. Are points A and D collinear? Yes, two points are always collinear.

The Triangle Inequality: For any three points, there are only two possibilies:

1. They are collinear. One point must be between the other two. Two of the distances must add to the third.

2. They are noncollinear. The three points will form a triangle. Any two of the lengths will add to be larger than the third.

Example 2: If AB=12, BC=13, and AC=25, can we determine if the points are collinear?

Yes, the lengths 12 and 13 add to 25, therefore the points are collinear. How to Interpret a Diagram: You Should Assume: You Should Not Assume: Straight lines and angles Right Angles Collinearity of points Congruent Segments Betweenness of points Congruent Angles Relative position of points Relative sizes of segments and angles

Example 3: Use the diagram to the right to answer the following questions. a. Is A BCD? A C

No b. Name 3 collinear points. B ABD,, c. Name two straight angles. D ABD, EBC E d. Name a point between A and D. B e. Area any segments in the diagram congruent?

No, we cannot assume congruent segments. congruent segments must be marked.