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1.1 Points, Lines and Planes

1.1 Points, Lines and Planes

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1.1 Points, Lines and Planes

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  1. 1.1 Points, Lines and Planes

  2. Undefined Terms • There are three undefined terms in Geometry. • They are Points, Lines and Planes. • They are considered undefined because they have only been explained using examples and descriptions.

  3. Points • Points are simply locations. • Drawn as a “dot.” • Named by using a Capital Letter • No size or shape. • Verbally you say “Point P” P

  4. B A Line l • A line is a collection of an infinite number of points (named or un-named). • Points that lie on the line are called Collinear. • Collinear Points are points that are on the same line. • Draw a line with arrows on each end to signify that it is infinite in both directions. • Name by either two points on the line or lower case “script letter”

  5. B A Line (Continued) • A line has only one dimension (length). • It has no width or depth. • Postulate –There exists exactly one line through two points. • To plot a point on a number line, you’ll need only one number. l

  6. K R F Plane • A plane is a flat surface made up of an infinite number of points. • Points that lie on the same plane are said to be Coplanar. • Planes are named by using a capital, script letter or three non-collinear points. Plane RFK Plane P P

  7. Plane (Continued) • Although a plane looks like it is a quadrilateral, it is in fact infinitely long and wide. • Planes (Coordinate Plane) have two dimensions – so you need two numbers to plot a point. P(x,y)

  8. Space • Space is a boundless, three dimensional set of all points. Space can contain, points, lines and planes. • In chapter 13 you will see that you’ll need three numbers to plot a point in space. P(x,y,z)

  9. Describing What you see! • There are key terms such as: • Lies in, • Contains, • Passes through, • Intersection, • See Pg 12.

  10. 1.2 Linear Measure and Precision

  11. B A Introduction • Lines are infinitely long. • There are portions of lines that are finite. In other words, they have a length. • The portion of a line that is finite is called a Line Segment. • A line segment or segment has two distinct end points.

  12. Betweenness • Betweenness of points is the relationships among three collinear points. • We can say B is between A and C and you should think of this picture. C A B Notice that B is between but not in exact middle.

  13. L M N Example Find the length of LN or LN=? From this picture we can always write this equation: LM + MN = LN. So, if LM = 3 and MN = 5, we can say that LN = 8. What if LM = 2y, MN = 21 and LN = 3y+1? Then we can write….. 2y + 21 = 3y + 1 From this equation we can solve for y and substitute that value to find LN.

  14. Congruence of Segments • Segments can be Congruent if they have the same measurement. • We have a special symbol for congruent. It is an equal sign with a squiggly line above it. Hint: Shapes can be congruent, measurements can only be equal. So if you’re talking about a shape, you say congruent or not congruent!

  15. Congruence • Congruence can not be assumed! • Don’t think, that just because it looks like the same length, it is. • Short cut… we can use congruent marks to show that segments are congruent. C Q A P

  16. Precision (H) • The precision of a measurement depends on the smallest unit of measure available on the measuring tool. • The precision will always be ½ the smallest unit of measure of the measuring device.

  17. 1 2 3 4 5 6 Precision (H) • Here to find the length we would have to say it is four units long b/c it is closer to 4 than 5. • The precision of this measuring device is ½ the smallest unit of measure, 1”, or the precision is ± 1/2. • We can say the measurement is 4 ± 1/2 • So the segment could be as small as 3 ½” or as big as 4 ½” and still be called 4”.

  18. 1 2 3 4 5 6 Precision (Con’t) • Here we have the same segment but a different, more accurate measuring device. • The units are broken down into ¼’s. The segment is closer to 4 ¼ than 4 ½. • The precision is ½ of ¼, or 1/8th. • So the length is 4 ¼ ± 1/8th. Smallest 4 1/8thLargest 4 3/8th.

  19. 1.3 Distance and Midpoints

  20. Distance • The coordinates of the two endpoints of a line segment can be used to find the length of the segment. • The length from A to B is the same as it is from B to A. • Thus AB = BA (This stands for the measurement of the segment) • Distance (length) can never be negative.

  21. A B C Midpoint • Definition - The midpoint of a segment is the point ½ way between the endpoints of the segment. • If B is the Midpoint (MP) of then, AB = BC. • The midpoint is a location, so it can be positive or negative depending on where it is.

  22. One Dimensional C B D A -3 -2 -1 0 1 2 3 4 • If point A was at -3 and point B was at 2, then AB=5 b/c the formula for AB = |A – B| • The MP formula is (A+B)/2 (-3+2)/2 = -1/2 • What if point C was at -2 and D was at 4, • what is CD? CD = |4 – (-2)| or | -2 – 4| = 6 MP is (4 + (-2))/2 = 1

  23. Two Dimensional • We designate points on a plane using ordered pair P(x,y). • We plot them on the Cartesian Coordinate plane just as you did in Alg I. • Again, distances can not be negative because lengths are not negative. • Midpoints can be either positive or negative b/c it is simply a location.

  24. d=√(8 – 2)2 + (10 – 5)2 = √36 + 25 = √61 Distance (Shortcut) (8, 10) 5 Find the distance between these two points. 6 (2, 5) Or use the Pythagorean theorem. Create a right triangle. d2 = 62 + 52 = 36 + 25 = 61 so d = √61

  25. 1.4 Angle Measure

  26. Z X Another Portion of a Line • We already talked about segments, now let us talk about Rays. • A ray is a portion of a line that has only one end point. It is infinite in the other direction. • A ray is named by using the end point and any other point on the ray.

  27. Opposite Rays • If you chose a point on a line, that point determines exactly two rays called Opposite Rays. • These two opposite rays form a line and are said to be collinear rays. C A B

  28. C E D Angles • Angles are created by two non-collinear rays that share a common end point. <CED or <DEC • Angles are named by using one letter from one side, the vertex angle, and one letter from the other side. • An angle consists of two sides which are rays and a vertex which is a point.

  29. C E D Interior vs. Exterior Exterior Interior Exterior Exterior

  30. Classifications of Angles • Right Angle – An angle with a measurement of exactly 90° m<ABC=90° • Acute Angle – An angle with a measurement more than 0° but less than 90° 0° < m<ABC < 90° • Obtuse Angle – An angle with a measurement more than 90° but less than 180° 90° < m<ABC < 180°

  31. C A D 25° 25° G E Congruence of Angles • Angles with the same measurement are said to be congruent. • m<ACE = 25° and m<DCG = 25°… since the two angles have the same measurement we can say that they’re congruent.

  32. A D H Angle Bisector • An angle bisector is a Ray that divides an angle into two congruent angles. P • If is an angle bisector…. • Then <ADP is congruent to <PDH.

  33. 1.5 Angle Relationships Angle Pairs

  34. A C D B Adjacent 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. <ABC and <CBD are Adjacent Angles. They don’t have to be equal. Common Side? Common Vertex? B No Common Interior Point?

  35. A C B E D Vertical Angles • Vertical Angles – Are two non-adjacent angles formed by intersecting lines. • Two Intersecting Lines? <ABD and <CBE are non-adjacent angles formed by intersecting lines. They are Vertical Pair. What else? <ABC and <DBE are also Vertical Pair.

  36. P L M N Linear Pair • Linear Pair – Is a pair of adjacent angles whose non-common sides are opposite rays. • Are <LMP and PMN are Adjacent? Yes! • Are Ray’s ML and MN the Non-Common Sides? Yes! • Are Ray’s ML and MN Opposite Rays? Yes! <LMP and <PMN are Linear Pair!

  37. 1 25° 65° 2 Complementary Angles • Complementary Angles – Are two angles whose measures have a sum of 90° • Do you see the word Adjacentin the definition? No! 1 2 <1 and <2 are Comp.

  38. P L M N 1 25° 2 155° Supplementary Angles • Supplementary Angles – Are two angles whose measures have a sum of 180° • Do you see the word Adjacent in the definition? No! 1 2 <1 and <2 are Supp.

  39. Perpendicular Lines • Perpendicular Lines intersect to form four right angles. • Perpendicular Lines intersect to form congruent, adjacent angles. • Segments and rays can be perpendicular to lines or to other line segments or rays. • The right angle symbol indicates that the lines are perpendicular.

  40. Assumptions • Things that can be assumed. • Coplanar, Intersections, Collinear, Adjacent, Linear Pair and Supplementary • Things that can not be assumed. • Congruence, Parallel, Perpendicular, Equal, Not Equal, Comparison.

  41. 1.6 Polygons

  42. Polygon • Polygon – A closed figure whose sides are all segments and they only intersect at the end points of the segments. • Polygons are named by using consecutive points at the vertices. • Example – A triangle with points of A, B and C is named ΔABC.

  43. Concave vs. Convex • Concave – A polygon is concave when at least one line that contains one of the sides passes through the interior. • Convex – A polygon is convex when none of the lines that contains sides passes through the interior. Concave Convex

  44. Classification by Sides • Polygons are classified by the number of sides it has. • 3 – Triangle 4 – Quadrilateral • 5 – Pentagon 6 – Hexagon • 7 – Heptagon 8 – Octagon • 9 – Nonagon 10 – Decagon • 11 – Undecagon 12 – Dodecagon • Any polygon more than 12 – then “N-Gon. Example 24 sides is a 24-gon.

  45. Regular Polygon • Regular Polygon – Is a polygon that is equilateral (all sides the same length), equiangular (all angles the same measurement) and convex. • Examples: • Triangles – Equilateral Triangle • Quadrilateral - Square

  46. Perimeter • Perimeter – The sum of the lengths of all the sides of the polygon. • May have to do distance formula for coordinate geometry problem. • See example #3.