1. Geometry Ms. Toney

2. Orientation • Student Information Sheet • Classroom Rules • Classroom Policies • Calendar

3. What you will learn today: • Identify and model points, lines, and planes. • Identify collinear and coplanar points and intersecting lines and planes in space. • Measure segments and determine accuracy of measurement. • Compute with measures.

4. Geometry in the Real – World? • Where are there points, lines, and/or planes in this classroom? • What about outside the classroom, like in nature?

5. Some Important Definitions • Point • A location • Drawn as a dot • Named by a capital letter • Has no shape and no size • Example:

6. Some Important Definitions • Line • Made up of points, has no thickness or width • Drawn with arrowhead at each end • Named by the letters representing two points on the line or a lower case script letter • There is exactly one line through any two points • Example: • Collinear • Points on the same line

7. Some Important Definitions • Plane • A flat surface made up of points • Has no depth and extends infinitely in all directions • Drawn as a shaded figure • Named by a capital script letter or by the letter naming the three noncollinear points • There is exactly one plane through any three noncollinear points • Points are often used to name lines and planes • Example: • Coplanar • Points that lie on the same plane

8. Example • Use the figure to name each of the following • A line containing point D • A plane containing point B

9. You Do It • Use the following figure to name each of the following • A line containing point K • A plane containing point L

10. Real – World Examples

11. Undefined Terms • Point, line and plane are undefined terms • Have only been explained using examples and descriptions • We can still use these to define other geometric terms and properties • Two lines intersect at a point • Lines can intersect planes • Planes can also intersect each other

12. Example • Draw and label a figure for each relationship • Lines GH and JK intersect at L for G(-1, -3), H(2, 3), J(-3, 2), and K(2, -3) on a coordinate plane. Point M is coplanar with these point but not collinear with lines GH or JK. • Line TU lies in a plane Q and contains point R

13. Your Turn • Draw and label a figure for each relationship • Line QR on a coordinate plane contains Q(-2, 3) and R(4, -4). Add point T so that T is collinear with these points • Plane R containing lines AB and DE intersect at point Add point C on plane R so that it is not collinear with lines AB or DE

14. Space • A boundless, three – dimensional set of all points • Can contain lines and planes

15. Example • How many planes appear in this figure? • Name three points that are collinear. • Are points G, A, B, and E coplanar? Explain.

16. Example • How many planes appear in this figure? • Name three points that are collinear. • Are points A, B, C, • and D coplanar? Explain.

17. Example • How many planes appear in this figure? • Name three points that are collinear. • Are points X, Y, Z, and P coplanar? Explain. • At what point do lines PR and TZ intersect?

18. Units of Measure • When you see these signs, what unit of measure do you believe is being used?

19. Units of Measure • Actually in Australia, the unit of measure is kilometers. • Units of measure give us points of reference when evaluating the sizes of objects.

20. Measure Line Segments • Line Segment • Also called a segment • Can be measured because it has two endpoints • Named: • The length or measure of is AB. • The length of a segment is only as precise as the smallest unit on the measuring device.

21. Example • Find AC. • Find DE. • Find y and PQ if P is between Q and R, PQ = 2y, QR = 3y + 1, and PR = 21.

22. Your Turn • Find LM. • Find XZ. • Find x and ST if T is between S and U, ST = 7x, SU = 45, and TU = 5x – 3.

23. More Terms • Congruent • When two segments have the same measure • Segments and angles are congruent • Distant and measures are equal

24. Classwork • Textbook • Page 9 • Problems 3 – 10 (all) • Page 16 • Problems 3 – 10 (all)

25. Homework • Textbook • Page 9 - 10 • 13 – 18, 23, 24, 30 – 34 • Page 17 • 12 – 15, 22, 26, 30, 32, 34, and 36

27. What you will learn today: • Find the distance between two points. • Find the midpoint of a segment. • Measure and classify angles. • Identify and use congruent angles and the bisector of an angle.

28. Distance • Is always positive • Because you use whole numbers • Ways to find distance: • Number line • Pythagorean Theorem • c2 = a2 + b2 • Distance Formula

29. Example • Use the number lines to find the following:

30. Example

31. Example

32. Your Turn

33. Midpoint • The point halfway between the endpoints of a segment • If B is the midpoint of the AB = BC • Two formulas: • Number Line • Coordinate Plane

34. Example • 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). • A is an endpoint and B is the midpoint located at A(3, 4) and B(-2, 1). Find the other endpoint C.

35. Your Turn • Find coordinates of D if E(-6, 4) is the midpoint of and F has coordinates (-5, -3). • What is the measure of if Q is the midpoint of ?

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

37. Angle Measure • Ray: • Part of a line • Has one endpoint and extends indefinitely in one direction • Named stating the endpoint first and then any other point on the ray

38. Angle Formed by two noncollinear rays that have a common endpoint Rays are called sides of the angle Common endpoint is the vertex More Terms

39. An angle divides a plane into three distinct parts Points A, D, and E lie on the angle Points C and B lie in the interior of the angle Points F and G lie in the exterior of the angle

40. Example • Name all the angles that have W as a vertex. • Name the sides of angle 1. • Write another name for angle WYZ.

42. Types of angles • Right Angle • Measurement of A = 90°

43. Types of Angles • Acute Angle • Measurement of B is less than 90°

44. Types of angles • Obtuse angles • Measurement of C is less than 180° and greater than 90°

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

46. Example • Wall stickers of standard shapes are often used to provide a stimulating environment for a young child’s room. A five – pointed star sticker is shown. Find and , , and

47. You Do It • A trellis is often used to provide a frame for vining plants. Some of the angles formed by the slats of the trellis are congruent angles. If

48. Bisectors • Segment Bisector • A segment, line or plane that intersects a segment at its midpoint • Angle Bisector • A ray that divides an angle into two congruent angles

49. Classwork • Textbook • Page 25 • 3 - 11 • Page 33 • 4 – 10 (all)

50. Homework • Workbook • Page 4 • 1 – 17 (all)