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Understanding Parallel Lines, Planes, and Skew Lines in Geometry

This chapter explores the concepts of parallel lines and parallel planes, which exist in the same plane without intersecting, and discusses skew lines that are not coplanar and never intersect. Examples of parallel planes (RSTV and XWZY; RSWX and TVYZ; STXY and WRVZ) illustrate their properties. A transversal line intersecting two lines creates various angle relationships, including corresponding, alternate exterior, alternate interior, and consecutive interior angles. Each type of angle relationship is clarified with specific examples and their congruence properties.

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Understanding Parallel Lines, Planes, and Skew Lines in Geometry

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  1. Geometry Journal Chapter 3By: Alejandro Gonzalez

  2. Describe parallel lines and parallel planes. Include a discussion of skew lines. Give at least 3 examples. • Parallel lines: Lines in the same plane that don't intersect. • Parallel Planes: Different planes that don't intersect. Planes RSTV and XWZY are parallel Planes RSWX and TVYZ are parallel Planes STXY and WRVZ • Skew lines: Are not coplanar lines that means that lines never intersect and are in different planes.

  3. Describe what a transversal is. Give at least 3 examples. • A transversal is a line that intersects two lines at two different points.

  4. Corresponding, alternate exterior, alternate interior and consecutive interior angles. Give an example of each. • Corresponding: In a transversal, the pair of angles that lay on the same side of the transversal, and on the same side of the other two lines. • Alternate Exterior: In a transversal, the pair of angles that are on the opposite sides of it, and outside the other two lines. • Alternate Interior: In a transversal, the pair of nonadjacent angles that are on the opposite sides of the transversal and between the two other lines. • Consecutive Interior: This are the same as Same side interior angles. There are the angles that are on the same side of the transversal and between the two lines.

  5. Examples: 2 1 Corresponding: <1 and <5 Alternate Exterior: <1 and <8 Alternate Interior: <4 and <5 Consecutive: <3 and <5 4 3 6 5 8 7

  6. Corresponding angle postulate and converse. 2 1 • When two lines are cut by a transversal the corresponding angles are congruent. <1≅<5 <2≅<6 <3≅<7 <4≅<8 4 3 6 5 8 7

  7. Alternate Interior Angles Theorem 2 1 • When the two parallel lines are cut by a transversal the alternate interior angles are congruent. <3≅<6 <4≅<5 4 3 6 5 8 7

  8. Same Side interior angle theorem 2 1 • When the two parallel lines are cut by a transversal the same side interior angles are congruent. <3≅<5 <4≅<6 4 3 6 5 8 7

  9. Alternate Exterior Angles Theorem 2 1 • When two parallel lines are cut by the transversal the alternate exterior angles are congruent <1≅<8 <2≅<7 4 3 6 5 8 7

  10. Perpendicular Transversal Theorem • In a plane, If a line is perpendicular to one of two parallel lines, then it is perpendicular to the other line also.

  11. The Transitive Property in parallel and perpendicular lines • Parallel: If two lines are parallel to a third line then the two lines are parallel to each other. • Perpendicular: If two lines are perpendicular to a third line then they are parallel to each other.

  12. Slope • To find the slope of a line you need to use the formula EX 1: (2 , 1) , (4 , 5) m= ( 5 - 1 ) / (4 - 2) = 4 / 2 = 2 EX 2: (-1 , 0) , (3 , -5) m= ( -5 - 0 ) / ( 3 - (-1) ) = -5 / 4 EX 3: (2 , 1) , (-3 , 1) m = ( 1 - 1 ) / ( -3 - 2 ) = 0 EX 4: (-1 , 2) , (-1 ,- 5) m = ( -5 - 2 ) / ( -1 - (-1)

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