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3-1 Lines and Angles

3-1 Lines and Angles. Today you will learn to identify relationships between figures in space. What would you call two lines which do not intersect?. Parallel. A solid arrow placed on two lines of a diagram indicate the lines are parallel. The symbol || is used to indicate parallel lines.

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3-1 Lines and Angles

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  1. 3-1 Lines and Angles Today you will learn to identify relationships between figures in space.

  2. What would you call two lines which do not intersect? Parallel A solid arrow placed on two lines of a diagram indicate the lines are parallel. The symbol || is used to indicate parallel lines. AB || CD

  3. A slash through the parallel symbol || indicates the lines are not parallel. AB || CD

  4. Skew Lines Two lines are skew if they are not in the same plane and do not intersect. AB does not intersect CD . Since the lines are not in the same plane, they are skew lines.

  5. Vocabulary • Parallel lines are coplanar lines that do not intersect. • Lines in different planes that do not intersect are skew. • Parallel planes are planes that do not intersect. • Perpendicular lines – lines that intersect and form right angles

  6. LM ||QR KN and PQ NS  SP Identifying Types of Lines and Planes Identify each of the following. A. a pair of parallel segments B. a pair of skew segments C. a pair of perpendicular segments D. a pair of parallel planes plane NMR|| plane KLQ

  7. BF || EJ BF and DE are skew. BF  FJ Identify each of the following. a. a pair of parallel segments b. a pair of skew segments c. a pair of perpendicular segments d. a pair of parallel planes plane FJH || plane BCD

  8. Think of each segment in the diagram. Which appear to fit the description? Parallel to AB and contains D Perpendicular to AB and contains D Skew to AB and contains D Name the plane(s) that contains D and appear to be parallel to plane ABE Identifying relationships in space B C D A F G E H

  9. Parallel Postulate • If there is a line and a point not on the line, then there is exactly one line through the point parallel to the given line. P l

  10. Perpendicular Postulate • If there is a line and a point not on the line, then there is exactly one line through the given point perpendicular to the given line. P l

  11. Exit Ticket List the plane(s) parallel to plane CDE. Homework: WB p. 59 #1-15

  12. 3-1 Lines and Angles Today you will learn to identify angles formed by parallel lines and a transversal.

  13. Transversals • If you have 2 coplanar lines that are intersected by a third (called a transversal), special angle pairs are formed. Transversal

  14. Interior Angles • Angles between the two lines are interior angles.

  15. Exterior Angles • Angles outside of the two lines are exterior angles.

  16. Same Side Angles • Angles on the same side of the transversal are same side angles.

  17. Alternate Angles • Angles on the opposite side of the transversal are alternate angles.

  18. Combined Angles • The types of angles mentioned previously are only useful to us in certain combinations: • Alternate interior • Same side interior • Alternate exterior • Same side exterior (rarely used)

  19. Corresponding Angles • If two angles occupy the same relative position at each of the points of intersection, they are corresponding angles. • Which pairs of angles are corresponding? 1 2 3 4 6 5 8 7

  20. Parallel Lines & Transversals • Parallel lines are lines in the same plane that do not intersect. • When a transversal intersects parallel lines, the special angles pairs take on certain properties. a b • These arrows are how parallel lines are marked on a diagram.

  21. Parallel Lines Cut by a Transversal • Get into pairs. On a sheet of paper, draw two parallel lines (either both horizontal or vertical). • Now use your ruler to draw a transversal that intersects both parallel lines. 2 1 Label these pairs of angles: 1 & 5 4 & 6 2 & 8 4 3 5 6 8 7

  22. Parallel Postulate (Remember, a postulate is something we accept as true without proof.) • If parallel lines are intersected by a transversal, then the corresponding angles are congruent. • In other words, if a ll b, then 1  2. a 1 b 2

  23. Parallel Lines & Transversal Parallel lines and transversals form special angles: • Corresponding angles are congruent • Alternate interior angles are congruent • Same side interior angles are supplementary • Alternate exterior angles are congruent

  24. A Special Case • If there are a pair of parallel lines, and a transversal is perpendicular to one of them, then it is perpendicular to the other. • If a ll b and a  t, thenb  t. t a b

  25. Be Able to Name Special Pairs of Angles: • Alternate Interior Angles • Corresponding Angles • Alternate Exterior • Same-Side Interior Angles 2 1 4 3 6 5 7 8 Be Able to State the Relationship Between Any Two Angles: • Congruent Angles • Supplementary Angles

  26. Exit Ticket • What three types of angles are congruent if lines are parallel and cut by a transversal? • Homework – WB p. 60 (ALL)

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