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3-1 and 3-2: Parallel Lines and Transversals

3-1 and 3-2: Parallel Lines and Transversals. Mr. Schaab’s Geometry Class Our Lady of Providence Jr.- Sr . High School 2014-2015. Identifying Pairs of Lines. Two lines are: Parallel if they do not intersect and are coplanar . Skew if they do not intersect and are not coplanar .

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3-1 and 3-2: Parallel Lines and Transversals

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  1. 3-1 and 3-2: Parallel Lines and Transversals Mr. Schaab’s Geometry Class Our Lady of Providence Jr.-Sr. High School 2014-2015

  2. Identifying Pairs of Lines • Two lines are: • Parallel if they do not intersect and arecoplanar. • Skew if they do not intersect and are not coplanar. • Perpendicular if they intersect to form right angles. (Note: ALL intersecting lines are coplanar!)

  3. Identifying Pairs of Lines - Example In the cube below, identify the following: • A pair of perpendicular lines: • Line PS and Line PW • A pair of parallel lines: • Line PW and Line QX • Line PW and Line RY • Line QX and Line RY • A pair of skew lines: • Line PS and Line QX • Line PS and Line RY

  4. Identifying Pairs of Planes - Example In the cube below, identify the following: • A pair of perpendicular planes: • Plane SRY and Plane PWZ • Plane PQX and Plane QXY • A pair of Parallel planes: • Plane PQX and Plane SRY • Plane PWZ and Plane QRY • Plane PQR and WXY • Pair of skew planes: • None!

  5. Angles and Transversals • Transversal – a line that intersects two or more coplanarlines at different points. In the diagram on the right, line tis a the transversal of lines L1andL2. A transversal that intersects two lines forms 8 angles, all of which have special relationships.

  6. Angle Relationships • Corresponding Angles • Two angles that are in matching locations on different intersections. • ∠1 and ∠5 are corresponding angles. 1 5

  7. Angle Relationships • Alternate Interior Angles • Two angles that lie between the two lines and on opposite sides of the transversal. • ∠4 and ∠5 are alternate interior angles. 4 5

  8. Angle Relationships • Alternate Exterior Angles • Two angles that lie outside the two lines and on opposite sides of the transversal. • ∠2 and ∠7 are alternate interior angles. 2 7

  9. Angle Relationships • Consecutive Interior Angles • Two angles that lie between the two lines and on the same side of the transversal. These are also called “Same-side interior angles.” • ∠3 and ∠5 are consecutive interior angles. 3 5

  10. Angles and Transversals - Example Identify all pairs of angles of the given type: • Corresponding: • ∠1 & ∠5, ∠2 & ∠6, ∠3 & ∠7, ∠4 & ∠8 • Alternate Interior: • ∠2 & ∠7, ∠4 & ∠5 • Alternate Exterior: • ∠3 & ∠6, ∠1 & ∠8 • Consecutive Interior: • ∠4 & ∠7, ∠2 & ∠5 6 5 8 7 2 1 4 3

  11. Parallel Lines and Transversals • Corresponding Angles Postulate: • If two parallel lines are cut by a transversal, then all pairs of corresponding angles are congruent. • ∠1 ≅ ∠5 1 5

  12. Parallel Lines and Transversals • Alternate Interior Angles Theorem: • If two parallel lines are cut by a transversal, then all pairs of alternate interior angles are congruent. • ∠4 ≅ ∠5 4 5

  13. Parallel Lines and Transversals • Alternate Exterior Angles Theorem: • If two parallel lines are cut by a transversal, then all pairs of alternate exterior angles are congruent. • ∠1 ≅ ∠8 1 8

  14. Parallel Lines and Transversals • Consecutive Interior Angles Theorem: • If two parallel lines are cut by a transversal, then all pairs of consecutive interior angles are supplementary. • m∠3 + m∠5 = 180° 3 5

  15. Parallel Lines and Transversals • If you’re angle 3, then you have a lot of relationships! The other angles must really like you. • ∠3& ∠1 – Linear Pair (supplementary) • ∠3& ∠2 – Vertical Angles (congruent) • ∠3& ∠4 – Linear Pair (supplementary) • ∠3& ∠5 – Consecutive Interior Angles (supplementary) • ∠3& ∠6 – Alternate Interior Angles (congruent) • ∠3& ∠7 – Corresponding Angles (congruent) • ∠3& ∠8 – No relationship (∠8 is a jerk.) 1 2 3 4 5 6 7 8

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