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3-1 3-2 3-3 Parallel Lines and Angles

3-1 3-2 3-3 Parallel Lines and Angles. Objectives Define transversal and the angles associated with a transversal State and apply the properties of angles associated with a transversal that cuts a pair of parallel lines. 3-1 Definition of transversal. Parallel Lines. b . l . 2. 1. 2. 1.

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3-1 3-2 3-3 Parallel Lines and Angles

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  1. 3-1 3-2 3-3 Parallel Lines and Angles • Objectives • Define transversal and the angles associated with a transversal • State and apply the properties of angles associated with a transversal that cuts a pair of parallel lines

  2. 3-1 Definition of transversal Parallel Lines b l 2 1 2 1 3 4 4 3 m c t 6 5 6 5 8 7 7 8 t is a transversal for l and m. • A transversal is a line that intersects two coplanar lines at two distinct points Nonparallel Lines r r is a transversal for b and c.

  3. 3-1 Angles associated with transversals b 2 1 3 4 c 6 5 7 8 Interior angles lie between the two lines. Examples: ∠4, ∠6 Lines b and c Exterior angles lie outside the two lines. Examples: ∠1, ∠8 Alternate Interior angles are on the opposite sides of the transversal. Example: ∠4 and ∠6 Alternate Exterior angles are on the opposite sides of the transversal. Example: ∠2 and ∠8 r Same Side Interior angles are on the same side of the transversal. Example: ∠4 and ∠5 r is a transversal for b and c. Corresponding angles are on the same side of the transversal and on the same side of the lines cut by the transversal. Example: ∠2 and ∠6

  4. 3-1 Theorems/postulates for parallel lines Parallel Lines l 2 1 4 3 m t 6 5 8 7 t is a transversal for l and m. • If a transversal intersects two parallel lines, then • Corresponding angles are congruent (Corr. Angles Postulate) • Alternate interior angles are congruent (Alt. Int. Angles Thm) • Alternate exterior angles are congruent (Alt. Ext. Angles Thm) • Same side interior angles are supplementary (Same Side Int. Angles Thm) • Same side exterior angles are supplementary (Same Side Ext. Angles Thm)

  5. 3-2 Converses of previous slide • All converses of the previous slide are true • Example: Converse of the Corresponding Angle Postulate: • If two lines and a transversal form congruent corresponding angles, then the two lines are parallel • Example: Converse of the Alternate Interior Angles Theorem: • If two lines and a transversal form congruent alternate interior angles, then the two lines are parallel

  6. 3-3 Parallel and perpendicular lines • If two lines are parallel to the same line, then they are parallel to each other. • In a plane, if two lines are perpendicular to the same line, then they are parallel to each other. • In a plane, if a line is perpendicular to one of two parallel lines, then it is also perpendicular to the other.

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