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Lesson 1.3: Segments, Rays, and Distance

Lesson 1.3: Segments, Rays, and Distance. Pre-AP Geometry. Points, Lines, and Planes. Line Segment Two points (called the endpoints ) and all the points between them that are collinear with those two points Named line segment AB, AB, or BA line AB segment AB 

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Lesson 1.3: Segments, Rays, and Distance

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  1. Lesson 1.3: Segments, Rays, and Distance Pre-AP Geometry

  2. Points, Lines, and Planes Line Segment Two points (called the endpoints) and all the points between them that are collinear with those two points Named line segment AB, AB, or BA line AB segment AB  A B A B

  3. Length of a segment Length of BC is stated as BC. It is the distance between points B and C. On a number line, length of a segment is found by subtracting the coordinates of the endpoints. On a coordinate plane, length of a segment is found using the distance formula D =

  4. Examples Find the length between 5 and -3 on the number line Find the distance of segment AB if A(-3, 5) and B(2, -7)

  5. Postulates • Postulate: statement that is accepted without proof • Segment Addition Postulate • If B is between A and C, then AB + BC = AC • Ruler Postulate • The points on a line can be paired with the real numbers in such a way that any two points can have the coordinates 0 and 1 • Once a coordinate system has been chosen in this way, the distance between any two points equals the absolute value of the difference of their coordinates.

  6. Examples E F G EG = 7x + 3 EF = 3x + 8 FG = 2x + 1 Find x: Find EG: Find EF: Find FG:

  7. Segment Length terms Congruent- two objects that have the same size and shape. We use the symbol ≅ to show that two objects are congruent. Congruent segments- two segments with equal lengths. Example: DE ≅ FG Midpoint of a segment: a point that divides a segment into two congruent segments. Midpoint formula: M = ( ) Segment bisector: A line, segment, ray, or plane which intersects a segment at its midpoint.

  8. Points, Lines, and Planes Ray Part of a line that starts at a point and extends infinitely in one direction. Initial Point Starting point for a ray. Ray CD, or CD, is part of CD that contains point C and all points on line CD that are on the same side as of C as D “It begins at C and goes through D and on forever”

  9. Points, Lines, and Planes Opposite Rays If C is between A and B, then CA and CB are opposite rays. Together they make a line. A C B

  10. Lesson 1.4: Angles

  11. Parts of an angle vertex sides Sides of an angle are made up of rays The rays meet at a point called the vertex

  12. Naming an angle G I 1 H An angle can be named by the vertex, by the 3 points on the angle: the side, the vertex and the other side, or a number inside the angle. The angle can be named ∠GHI, ∠IHG, ∠H, or ∠1

  13. Classifying angles Acute angle: Angle measuring greater than 0° and less than 90°. Obtuse angle: Angle measuring greater than 90° and less than 180° Right angle: An angle measuring exactly 90° Straight angle: An angle measuring exactly 180°

  14. Angle Postulates • Protractor Postulate: On AB in a given plane, choose any point O between A and B. Consider OA and OB and all the rays that can be drawn from O on one side of AB. These rays can be paired with real numbers from 0 to 180 in a way such that: a. OA is paired with 0, and OB with 180 b. If OP is paired with x, and OQ with y, the m∠POQ = │x - y │ • Angle addition postulate: -If B lies on the interior of ∠AOC, then m ∠AOB + m∠BOC = m∠AOC -If ∠AOC is a straight angle, then m∠AOB+m ∠BOC = 180.

  15. Angle Vocabulary • Congruent Angles • Two angles with equal measures • Adjacent angles • Angles which share a vertex and a common side, but no common interior points • Angle bisector • A ray which divides an angle into two congruent, adjacent angles

  16. Congruence symbols and drawing conclusions Do not assume anything in geometry. Just because two segments look equal does not mean that they are.

  17. Postulates and Theorems Relating Points, Lines, and Planes Lesson 1.5 Pre-AP Geometry

  18. Postulates A point is defined by its location. A line contains at least two points. A plane contains at least three points not all in one line. Space contains at least four points not all in one plane.

  19. Postulates Through any two points there is exactly one line. Through any three points there is at least one plane and through any three non-collinear points there is exactly one plane. If two points are in a plane, then the line that contains the point is in that plane. If two planes intersect, then their intersection is a line.

  20. Theorem If two lines intersect, then they intersect in exactly one point.

  21. Theorem Through a line and a point not in the line there is exactly one plane.

  22. Theorem If two lines intersect, then exactly one plane contains the lines.

  23. Review Quiz – True or False A given triangle can lie in more than one plane. Any two points are collinear. Two planes can intersect in only one point. Two lines can intersect in two points.

  24. Review Quiz – True or False A given triangle can lie in more than one plane. False. Through a line and a point not in the line there is exactly one plane. Any two points are collinear. True. Two planes can intersect in only one point. False. If two planes intersect, then they intersection is a line. Two lines can intersect in two points. False. If two lines intersect, then they intersect in exactly one point.

  25. Problem Set 1.5 Written Exercises p.25: # 1 – 13 (odd), 14 - 20

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