Unit 1

1. Unit 1 Describe and Identify the three undefined terms, Understand Segment Relationships and Angle Relationships

2. Part 1 Definitions: Points, Lines, Planes and Segments

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

11. Midpoint / Segment Bisector • The midpoint of a segment is the point that divides the segment into two congruent segments • The Segment Bisector is a segment, line or ray that intersects another segment at its midpoint.

12. Example • Q is the Midpoint of PR, if PQ=6x-7 and QR=5x+1, find x, PQ, QR, and PR.

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

14. 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

15. Example Find the length XY in the figure shown.

16. Example • If S is between R and T and RS = 8y+4, ST = 4y+8, and RT = 15y – 9. Find y.

17. Part 3 Angles

18. 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.

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

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

21. 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

22. Example • If m<BAC = 155, find m<CAT and m<BAT

23. Example • <ABC is a straight angle, find x.

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

25. Example • Ray KM bisects <JKL, if m<JKL=72 what is the m<JKM?

26. Adjacent Angles • are two angles that lie in the same plane, have a common vertex, and a common side, but no common interior points

27. Vertical Angles • Two non-adjacent angles formed by two intersecting lines • Vertical Angles have the same measure and are congruent

28. Linear Pair • 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. Examples

34. Part 3 Polygons

35. Polygon • Closed figure whose sides are all segments. • To be a Polygon 2 things must be true • Sides have common endpoints and are not collinear • Sides intersect exactly two other sides

36. Naming a Polygon • The sides of each angle in a polygon are the sides of the polygon • The vertex of each angle is a vertex of the polygon • They are named using all the vertices in consecutive order

37. The number of sides determines the name of the polygon • 3 - Triangle • 4 - Quadrilateral • 5 - Pentagon • 6 - Hexagon • 7 - Heptagon • 8 - Octagon • 9 - Nonagon • 10 - Decagon • 12 - Dodecagon • Anything else …. N - gon (where n represents the number of sides)

38. Concave VS Convex

39. Regular Polygon • A regular polygon is a convex polygon whose sides are all congruent and whose angles are all congruent

40. Perimeter • The perimeter of a polygon is the sum of the lengths of its sides.

41. Perimeter of the Coordinate Plane • Find the perimeter of the triangle ABC with A(-5,1), B(-1,4), C(-6,-8)

42. Area • Area of a polygon is the number of square units it encloses

43. Circle

44. Unit 1 The End!