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1.5 – Basic Postulates in Geometry

Objectives

  • Apply knowledge of points, lines, and distance along a line to create precise definitions for an angle and line segment.
  • Use dimensions to precisely describe points, line segments, rays, lines, and angles.
  • Identify acute, obtuse, and right angles when given angle measures or diagrams.
  • Identify linear, supplementary, and complementary angle pairs in diagrams.
  • Define and apply congruence to segment midpoints and angle bisectors.
  • Use precise notation to distinguish between a line segment and its measure, and between congruent and equal figures.

 

Key Terms

  • Acute Angle – an angle that measures less than 90°. An acute angle is smaller than a right angle.
  • Adjacent Angles – angles that share a vertex and one side.
  • Angle – the object formed by two rays that share the same endpoint.
  • Angle Addition Postulate – if point C lies in the interior of AVB, then m AVC + m CVB = m AVB.
  • Angle Bisector – a ray that divides an angle into two angles of equal measure.
  • Complementary – having angle measures that add up to 90°. If two complementary angles are adjacent, they form a right angle.
  • Congruent – having the same size and shape. If polygons are congruent, their corresponding sides and angles are also congruent. The symbol means “congruent.”
  • Endpoint – a point at the end of a ray, either end of a line segment, or either end of an arc.
  • Linear Pair – a pair of adjacent angles whose measures add up to 180°. Linear pairs of angles are supplementary.
  • Line – the set of all points in a plane that are equidistant from two points.
  • Line Segment – a part of a line with endpoints at both ends. The symbol AB means “the line segment with endpoints A and B.” It is sometimes called a segment.
  • Midpoint – the point halfway between the endpoints of a line segment.
  • Point – the most basic object in geometry, used to mark and represent locations. Points have no length, width, or height.
  • Obtuse Angle – an angle with a measure greater than 90° but less than 180°.
  • Ray – a part of a line that starts at an endpoint and extends forever in one direction. The symbol means “the ray with endpoint A that passes through B.”
  • Right Angle – an angle that measures 90°. Right angles are often marked with a small square symbol. Perpendicular lines form right angles.
  • Segment Addition Postulate – a postulate stating that if AC + BC = AB, then point C is between points A and B.
  • Straight Angle – an angle whose sides form a line. The measure of a straight angle is 180°.
  • Supplementary – having angle measures that add up to 180°. If two supplementary angles are adjacent, they form a straight angle.
  • Vertex – a point at which rays or line segments meet to form an angle. The plural of vertexis vertices. The vertices of a polygon are the points at which the sides meet. The vertices of a polyhedron are the points at which at least three edges meet.
  • Zero Angle – an angle that has a measure of zero degrees and whose sides overlap to form a ray.


Notes

  • Segment Notation can be written 3 different ways
    • 1. Segment Notation (Sentence): The length of \overline{AB} is 3 inches.
      • The word length (above) is required for \overline{AB} to have a bar on top of the letters.
    • 2. Length Notation (Sentence): AB is 3 inches.
    • 3. Length Notation (Equation): AB = 3 inches.

 

  • Angle Notation
    1. Use the vertex only
    2. Use the vertex and two other points (vertex is in the middle)
    3. Use the number assigned to the angle

Geo A - 1.05 Angle Notation

 

  • A protractor is used to measure angles in degrees (Apex Pg 12).
    • To measure an angle with a protractor, follow these steps:
      • Step 1: Position the protractor’s red circle over the angle’s vertex.
      • Step 2: Align the “0” mark with one of the angle’s sides.
      • Step 3: See where the other side of the angle falls on the protractor.
        • This shows the difference between the rays’ directions, which is the angle’s measure.

Geo A - 1.05 Protractor

 

  • Angle Measures
    • A zero angle is a ray and measures 0°.
    • A straight angle is a straight line and measures 180°.
    • A right angle measures 90°.
    • An acute angle measures less than 90°.
    • An obtuse angle measures more than 90°.

 

  • m∠AVM means “the measure” of ∠AVM
  • The angle addition postulate states that if a point C is inside ∠AVB, then m∠AVC + m∠CVB = m∠AVB

Geo A - 1.05 Angle Addition Postulate

 

  • Congruent angles have the same angle measure

Geo A - 1.05 Congruent Angles

Geo A - 1.05 Congruency Comparison

Geo A - 1.05 Notation

 


Geo A - 1.05 Basic Postulates - 01 Geo A - 1.05 Basic Postulates - 02 Geo A - 1.05 Basic Postulates - 03

Geo A - 1.05 Basic Postulates - 04

 


Geo A 1.05 - Angles1 Geo A 1.05 - Angles2


 

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