Unit 1

1. Unit 1 Learning Outcomes 1: Describe and Identify the three undefined terms Learning Outcomes 2: Understand Angle Relationships

2. Part 1 Definitions: Points, Lines and Planes

3. Undefined Terms • Points, Line and Plane are all considered to be undefined terms. • This is because they can only be explained using examples and descriptions. • They can however be used to define other geometric terms and properties

4. Point • A location, has no shape or size • Label: • Line • A line is made up of infinite points and has no thickness or width, it will continue infinitely.There is exactly one line through two points. • Label: • Line Segment • Part of a line • Label: • Ray • A one sided line that starts at a specific point and will continue on forever in one direction. • Label:

5. Collinear • Points that lie on the same line are said to be collinear • Example: • Non-collinear • Points that are not on the same line are said to be non-collinear (must be three points … why?) • Example:

6. Plane • A flat surface made up of points, it has no depth and extends infinitely in all directions. There is exactly one plane through any three non-collinear points • Coplanar • Points that lie on the same plane are said to be coplanar • Non-Coplanar • Points that do not lie on the same plane are said to be non-coplanar

7. Intersect • The intersection of two things is the place they overlap when they cross. • When two lines intersect they create a point. • When two planes intersect they create a line.

8. Space • Space is boundless, three-dimensional set of all points. Space can contain lines and planes.

9. Practice Use the figure to give examples of the following: • Name two points. • Name two lines. • Name two segments. • Name two rays. • Name a line that does not contain point T. • Name a ray with point R as the endpoint. • Name a segment with points T and Q as its endpoints. • Name three collinear points. • Name three non-collinear points.

10. Part 2 Distance, Midpoint and Segments

11. Distance Between Two Points • Distance on a number line • PQ = or • Distance on coordinate plane • The distance d between two points with coordinates is given by

12. Examples • Example 1: • Find the distance between (1,5) and (-2,1) • Examples 2: • Find the distance between Point F and Point B

13. Congruent • When two segments have the same measure they are said to be congruent • Symbol: • Example:

14. Between • Point B is between point A and C if and only if A, B and C are collinear and

15. Midpoint • Midpoint • Halfway between the endpoints of the segment. If X is the MP of then

16. Finding The Midpoint • Number Line • The coordinates of the midpoint of a segment whose endpoints have coordinates a and b is • Coordinate Plane • The coordinates of midpoint of a segment whose endpoints have coordinates are

17. Examples • The coordinates on a number line of J and K are -12 and 16, respectively. Find the coordinate of the midpoint of • Find the coordinate of the midpoint of for G(8,-6) and H(-14,12).

18. Segment Bisector • A segment bisector is a segment, line or plane that intersects a segment at its midpoint.

19. Segment Addition Postulate • if B is between A and C, then AB + BC = AC • If AB + BC = AC, then B is between A and C

20. Part 3 Angles

21. Angle • An angle is formed by two non-collinear rays that have a common endpoint. The rays are called sides of the angle, the common endpoint is the vertex.

22. Kinds of angles • Right Angle • Acute Angle • Obtuse Angle • Straight Angle / Opposite Rays

23. Congruent Angles • Just like segments that have the same measure are congruent, so are angles that have the same measure.

24. Angle Bisector • A ray that divides an angle into two congruent angles is called an angle bisector.

25. Angle Addition Postulate • If R is in the interior of <PQS, then m<PQR + m<RQS = m<PQS • If m<PQR + m<RQS = m<PQS, then R is in the interior of <PQS

26. Measuring Angles • How to use a protractor. • 1.) Line up the base line with one ray of your angle. • 2.) Follow the base line out to zero, if you are at 180 switch the protractor around. • 3.) Trace to protractor up until you reach the second ray of your angle. • 4) The number your finger rests on is your angle measure.

27. Part 4 Angle Relationships

28. Pairs of Angles • Adjacent Angles - are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points • Vertical Angles-are two non-adjacent angles formed by two intersecting lines • Linear Pair - is a pair of adjacent angles who are also supplementary

29. Angle Relationships • Complementary Angles - Two angles whose measures have a sum of 90 • Supplementary Angles - are two angles whose measures have a sum of 180

30. Part 5 Angle Theorems

31. Theorem 2.3 • Supplement Theorem - • If two angles form a linear pair, then they are supplementary angles

32. Theorem 2.4 • Complement Theorem • If the non-common sides of two adjacent angles form a right angle, then the angles are complementary angles.

33. Theorem 2.6 • Angles supplementary to the same angle or to congruent angles are congruent

34. Theorem 2.7 • Angles complementary to the same angle or to congruent angles are congruent

35. Theorem 2.8 • Vertical Angles Theorem • If two angles are vertical, then they are congruent

36. Part 6 Perpendicular Lines and their theorems

37. Perpendicular Lines • Lines that form right angles are perpendicular • Perpendicular lines intersect to form 4 right angles • Perpendicular lines form congruent adjacent angles • Segments and rays can be perpendicular to lines or to other line segments or rays • The right angle symbol in a figure indicates that the lines are perpendicular.

38. Theorems • Theorem 2.9 - Perpendicular lines intersect to form four right angles • Theorem 2.10 - All right angles are congruent • Theorem 2.11 - Perpendicular lines form congruent adjacent angles

39. More Theorems • Theorem 2.12 - If two angles are congruent and supplementary, the each angle is a right angle • Theorem 2.13 - If two congruent angles form a linear pair, then they are right angles.

40. Unit 1 The End!