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Euclidean Geometry

Euclidean Geometry. Angles, Lines, Parallels. Definitions. Segment Two points and all points on the line between Ray A segment and all points on a line which includes the segment Angle at Vertex A single point with two rays. Definitions. Vertical angles

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Euclidean Geometry

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  1. Euclidean Geometry Angles, Lines, Parallels

  2. Definitions • Segment • Two points and all points on the line between • Ray • A segment and all points on a line which includes the segment • Angle at Vertex • A single point with two rays

  3. Definitions • Vertical angles • Two angles on the same point forming two lines • (i.e. vertical = “of one vertex”, not “upward”) • Supplementary Angles • Two angles sharing a line on opposite sides

  4. “Congruent” angles • Given two angles, are they same / identical? • Euclid • Put one angle on top of the other and it matches perfectly. • Modern • Use a measure for the angle, and compare the quantity.

  5. “Congruent” angles • Modern angular measures • Degrees(arbitrary) and Radians(from circle formula) • Range for measure of an angle is between0o to 180o or 0 to π radians. • Two angles are congruent if they have the same measure. • Measure of a right angle is 90o or ½π rad. • Two supplementary angles have a total measure of 180o or π rad.

  6. Definitions (2) • Degree measure • Given an angle ∠ABC, then the measure of the angle is m∠ABC • Length • Given a segment AB, the length is AB • Right angle • Angle as such its supplementary angle is congruent to it. • The supplementary is the same as itself.

  7. Definitions (2) • Perpendicular • Two lines where an angle at the intersection is a right angle. • Parallel • Two lines which do not intersect. • Other implications? • Bisector • For a segment AB, a bisector is the point C on it, as such AC is congruent to CB. • For an angle ∠BAC, a bisector is the ray AD such that ∠BAD and ∠DAC are congruent.

  8. Proving Propositions • Prop.9: To bisect a given rectilinear angle. • Prop.10 iswhen youbisect aline segment.

  9. Proving Propositions • Prop.11 To draw a straight perpendicular line, given a line and point on line. • Prop.12 is whenthe point is not onthe line.

  10. Proving Propositions • Prop.13 If a straight line stands on a straight line, then it makes either two right angles or angles whose sum equals two right angles.

  11. Proving Propositions • Prop.15 Two lines meet, make congruent vertical angles. • Prop.14 Two adjacent angles adding up to two right angles, form a line. • Prop.23 Recreate an angle on top of another line.

  12. Angles when Two Linesare Crossed • Given two lines L and M • Line T crosses L, M • Intersect pts A , A’ on L, M • Choose pts B , C on L • Choose pts B’ , C’ on M • Where are the • Interior angles • Exterior angles • Alternate angles • Corresponding angles

  13. Parallels • Prop.16 Exterior Angle Theorem • ∠DAB greater than ∠BCA • ∠DAB greater than ∠ABC • Prop.27 If a transversal crossing two lines gives alternate interior angles that are congruent, then the two lines are parallel.

  14. Parallels • Prop.28 If corresponding angles are congruent (alt. total of interior angles on one side equal two right angles) then the two transversed lines are parallel.

  15. Parallels • Prop.29 If N transverses parallel lines L and M, then • Alternate interior angles congruent • Corresponding angles congruent • Sum of (measures of) two interior angles on the same side is equal to two right angles

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