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1.2 – Points, Lines, and Planes Warm – Up, Do if you finished checking homework

1.2 – Points, Lines, and Planes Warm – Up, Do if you finished checking homework Find the slope of the line going through these two points. Because I know some of you forgot: (4,5) (-1,3) (-3,1) (2,-5) Does order matter? When? Why?.

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1.2 – Points, Lines, and Planes Warm – Up, Do if you finished checking homework

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  1. 1.2 – Points, Lines, and Planes • Warm – Up, Do if you finished checking homework Find the slope of the line going through these two points. Because I know some of you forgot: • (4,5) (-1,3) (-3,1) (2,-5) • Does order matter? When? Why?

  2. A definition uses known words to describe a new word. In geometry, are undefined terms. Point. Technically, it has no size, but we use a dot that has size to represent it. You use a capital letter to label it. Such as Point A All figures are made of points. This is a LINE. It goes both ways, forever without ending. Once again, it has no thickness, but we use a picture with thickness to describe it. Arrows on both ends say it goes on forever.

  3. PLANE, goes on forever, once again has no thickness. Even though it goes on forever, we usually use a parallelogram shape to draw it. A K I M To label it, a capital cursive letter can be used, or you can use three points that don’t line up (also known as non-collinear points)

  4. D U C K S Collinear points, points all in one line. Noncollinear points, points NOT all in one line. Coplanar points, points all in one plane. Noncoplanar points, points NOT all in one plane.

  5. R I T B F This is a ray This is a line segment, it is a segment, or part of a line T, R are ENDPOINTS

  6. l M N O OPPOSITE RAYS – are called opposite rays cuz N is between M and O.

  7. Name four coplanar points l P B Name three collinear points A D C What is the intersection of line l and plane P? E Q G I Which plane has points F,H,I? F J H

  8. Warm – Up, Do if you finished checking homework • Solve for y. Remember, that means make it ‘y=‘ 2x + 3y = 6 y – 3 = -2(x – 1) What does collinear mean? Why do two points HAVE to be collinear?

  9. Two or more geometric figures intersect if they have one or more points in common. The intersection of the figures is the set of points the figures have in common. Draw three noncollinear points A, B, C on plane P. Draw line l not on plane P going through point C.

  10. Draw three planes M, N, P meeting at point P. Draw three planes M, N, P meeting on line l. In 3-D, sometimes it helps to imagine a box, or look around the room (but not during a test)

  11. Four points, A, B, C, D, where A, B, and C are coplanar but not collinear on plane P, and D is noncoplanar • Three lines l, m, n meet at point P, where lines m, n are on plane P.

  12. B D F A C E 1.3 Segments and Their Measures Warm – Up: Find the distance between the points Where your ruler is doesn’t matter. Two points are the same distance apart no matter how you line up the ruler. Two find the distance between two points, you can just subtract the distance and take the absolute value. Like if on a number line, I have a points at Q and R, and they are different, to find the distance, all I have to do ___________ and then ____________

  13. Postulate \ Axiom – A rule that is accepted WITHOUT PROOF. Postulate 1 – Ruler Postulate The points on a line can be matched one to one with the real numbers. The real number that corresponds to a point is the coordinate of the point. The distance between points A and B, written as AB, is the absolute value of the difference between the coordinates of A and B. AB is also called the length of AB. A B -1 0 1

  14. AC A B C AB BC SEGMENT ADDITION POSTULATE If B is BETWEEN A and C, then AB + BC = AC. Also If AB + BC = AC, then B is between A and C Why does this only work if it’s ___________?

  15. B E O A X T

  16. DS = 30 DU = 5 KS = 7 UC = .5CK D U C K S UK = UC = DC = US = boardwork

  17. 5 5 H S T R Congruence is shown with marks. The marks say that they are the same size and shape Equals means they have equal length, number value. They are equivalent. Definition of congruent segments: Congruent segments have equal lengths Superposition:

  18. What is Pythagorean Theorem? What does a, b, and c represent? Looking at the graph, one way to represent the length of the horizontal leg is: (x2,y2) (x1,y1) Using the same logic, another way to represent the vertical leg is: Replace a and b with what we just found and solve for c

  19. (-5, -2) (4, 1) x1y1x2y2 Why do they use d instead of c? How come order doesn’t really matter for this formula? Why do you think they set it up this way?

  20. Find the distance between Mr. Kim and each food location. (0, 12) (8, 6) (0, 0) (16, 0) (8, 0) From where Mr. Kim starts, if he goes to In-N-Out, Der Veener, and Carl’s, and back to where he started, how far does he walk? What’s nice about finding distance when lines are horizontal or vertical?

  21. 1.4 – Angles and Their Measures • Warm – Up • Graph these lines. We will use them later. • x = 0 • y = 0 • y = x • x = 0 is also known as: • y = 0 is also known as:

  22. L A E N 1 G S Angles are formed by two rays with the same initial point. Two rays are called the sides. The initial point is called the vertex

  23. If two angles are congruent, their measures are equal. If the measure of two angles are equal, they are congruent D R 1 2 U E C X

  24. How to use a protractor to measure angles.

  25. Protractor Postulate A O B Consider a point A on one side of OB. The rays of the form OA can be matched one to one with the real numbers from 0 to 180. The measure of AOB is equal to the absolute value of the difference between the real numbers for OA and OB.

  26. Acute – Angle is between __ and __ degrees Right – Angle is exactly __ degrees Obtuse – Angle is between __ and ___ degrees 90 180 0 Straight – Angle is ___ degrees

  27. A point is in the _______ of an angle if it is between points that lie on each side of the angle. A points is in the _______ of an angle if it is not on the angle or its interior D U C Let’s look at the warm-up and identify angels, interior, and exterior points.

  28. Adjacent angles, share common side and vertex, but share NO interior points.

  29. C O B In the future with proofs, angle and segment addition postulates will be important in putting together and breaking apart angles. A

  30. Find the measure of the unknown angles, state if they are acute, right, or obtuse. 1 2 3 4 A D C B E F 1 76o

  31. Draw angle ABC that is 90o. Draw right angle DBF so that angle ABF and DBA is 45o and A is in the interior of angle DBF and F is in the interior of angle ABC.

  32. Draw a right angle KIM. Draw angle JIQ such that M is in the interior of angle JIQ and Q is in the interior of KIM and JIM is 30 degrees and MIQ is 60 degrees

  33. 1.5 – Segment and Angle Bisectors Warm – Up: What coordinate is in the MIDDLE of these two points? (-3, -2) (5, -1) x1y1x2y2 How did you find it? What’s another way to think of the ‘middle’ of two numbers?

  34. Find the midpoint. (5, -2) (3, 6) (-2, -1) (2, 5)

  35. D B A C E SEGMENT BISECTOR – A line, segment, or ray that INTERSECTS THE _____________________________________! The ___________ of a segment divides the segment into __________________parts.

  36. Given an endpoint and the midpoint, find the other endpoint. A is an endpoint, M is a midpoint A (5, -2) M (3, 6) B (x, y) A (2, 6) M (-1, 4) B (x, y)

  37. B A T R ANGLE BISECTOR – is a ray that divides an angle into two adjacent angles that are congruent.

  38. A A D D B B C C

  39. Constructing a perpendicular bisector. 1) Point on one end, arc up and down. 2) Switch ends and do the same 3) Draw line through intersection This is DIFFERENT from book (slightly).

  40. Bisect an angle 1) Draw an arc going across both sides of the angle. 2) Put point on one intersection, pencil on other, draw an arc so that it goes past at least the middle. 3) Flip it around and to the same. 4) Line from vertex to intersection.

  41. 1.7 – Introduction to Perimeter, Circumference, and Area Warm – Up: Things you should know from your past, fill in the blanks Square A = P = s Rectangle A = P = l w Perimeter of a triangle, add up the sides

  42. Circumference is the distance around the circle. (Like perimeter) C = πd = 2πr Area of a circle: A = πr2

  43. Find Perimeter\Circumference, and Area for each shape 13 cm 15 cm 5 ft 12 cm 14 cm 3 ft 3 in 6 ft

  44. Find the area and perimeter 12 cm 17 cm 8 cm

  45. Find the area of the figure described Find the area of a circle with diameter 10 m Find the area of a rectangle with base 4 ft and height 2 ft Find the area of a triangle with base 2 in and height 6 in Find the area of a square with perimeter 8 miles Write on board

  46. Finding Area

  47. Mr. Kim needs to make a moat around his castle. The radius of the outer circle is 50 feet, the radius of the inner circle is 40 feet. What is the area of his moat? How many square yards of flooring are needed to cover a room that is 18 ft by 21 ft?

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