Carnegie 1.1

1. Carnegie 1.1 Point, Line, Plane

2. Three Undefined Terms • Point • Line • Plane These three terms are called undefined terms; we can only describe and create mathematical models to represent them.

3. Point • A point is described as a location in space. A point has no size or shape but is often represented using a dot.

4. Line • A line is described as a straight continuous arrangement of an infinite number of points. A line has an infinite length, but no width. Arrowheads are used to indicate that a line extends infinitely in opposite directions.

5. More about Lines • Line AB can be written using symbols as  AB and is read as “line AB.” Collinear points are points that are located on the same line.

6. Plane • A plane is described as a flat surface. A plane has an infinite length and width but no depth. A plane extends infinitely in all directions. One model of a plane is the surface of a still body of water. Planes are determined by three points, but are usually named using one italic letter located near a corner of the plane as drawn.

7. Class Work • Complete Problems 1.1 and 1.3 before tomorrow.

8. Exit Ticket • Summarize what Point, Line, Plane are. • Draw an example of a point, line, and plane be sure to name them properly.

9. Ray • A ray is a portion of a line that begins with a single point and extends infinitely in one • direction. The endpoint of a ray is the single point where the ray begins. • A ray is named using two capital letters, the first representing the endpoint and • the second representing any other point on the ray. Ray AB can be written using • symbols as → AB and is read as “ray AB.”

10. Line Segment • A line segment is a portion of a line that includes two points and all of the collinear • points between the two points. The endpoints of a line segment are the points where the line segment begins and ends. • A line segment is named using two capital letters representing the two endpoints of the line segment. Line segment AB can be written using symbols as • ___ • AB and is read as “line segment AB.”

11. Congruent Line Segments • If two line segments have equal measure, then the line segments have the same length. Congruent line segments are two or more line segments of equal measure.

12. 1.2 All about Angles • Naming Angles • Classifying Angles • Duplicating Angles • Bisecting Angles

13. Naming Angles • An angle is formed by two rays that share a common endpoint. The angle symbol is . The sides of an angle are the two rays. The vertex of an angle is the common endpoint of the two rays.

14. Classifying Angles • An acute angle is an angle whose measure is greater than 0º, but less than 90º.

15. Right Angles • A right angle is an angle whose measure is equal to 90º. A square drawn at the vertex is used to indicate a right angle in geometric figures.

16. Obtuse Angles • An obtuse angle is an angle whose measure is greater than 90º, but less than 180º.

17. Straight Angle/Linear Angle • A straight angle is an angle whose measure is equal to 180º. The sides of a straight angle form a line.

18. Congruent • As with segments, use the congruent symbol, , between the geometric figure • angles, and the equal symbol, , between references to measures of angles.

19. Supplements and Complements • Two angles are supplementary angles if the sum of their angle measures is equal to 180º.

20. Complementary Angles • Two angles are Complementary angles if the sum of their angle measures is equal to 90º.

21. Perpendicular Relationships • Two lines, line segments, or rays are perpendicular if they intersect to form 90º angles. The perpendicular symbol is .

22. Adjacent Angles • Adjacent angles are two angles that share a common vertex and share a common side.

23. Linear Pairs • A linear pair of angles are two adjacent angles that have noncommon sides that form a line.

24. Vertical Angles • Vertical angles are two nonadjacent angles that are formed by two intersecting lines.

25. Class Work • Students will complete problems 1.1, and 1.2 before class tomorrow.

26. 1.3 Special Angles • Complements • Supplements • Midpoints • Perpendiculars • Perpendicular Bisectors

27. Supplements and Complements • Two angles are supplementary angles if the sum of their angle measures is equal to 180º.

28. Two angles are complementary angles if the sum of their angle measures is equal • to 90º.

29. Two lines, line segments, or rays are perpendicular if they intersect to • form 90º angles. The perpendicular symbol is .

30. The midpoint of a segment is a point that divides the segment into two congruent • segments, or two segments of equal measure. • P M Q • ___ • PQ has midpoint M. • A segment bisector is a line, line segment, or ray that divides the line segment into • two line segments of equal measure, or two congruent line segments. • A perpendicular bisector is a line, line segment, or ray that intersects the midpoint • of a line segment at a 90 degree angle.

31. Adjacent Angles • Adjacent angles are two angles that share a common vertex and share a • common side.

32. Linear Pairs • A linear pair of angles are two adjacent angles that have noncommon sides that • form a line.

33. Vertical Angles • Vertical angles are two nonadjacent angles that are formed by two • intersecting lines.

35. Your Assignment for the day. • Students will complete problems 1,5,6,7 from section 3, students will be prepared to share their answers with the class when we meet next.