Parallels 4.1 Parallel Lines and Planes

4.4 Proving Lines Parallel

4.3 Transversals and Corresponding Angles

4.2 Parallel Lines and Transversals

4.6 Equations of Lines

4.5 Slope

Parallel Lines and PlanesYou will learn to describe relationships among lines, parts of lines, and planes.In geometry, two lines in a plane that are always the same distance apart are ____________.parallel linesNo two parallel lines intersect, no matter how far you extend them.

Parallel Lines and Planesintersect

Definition ofParallel LinesTwo lines are parallel iff they are in the same plane and do not ________.

Parallel Lines and PlanesPlanes can also be parallel.The shelves of a bookcase are examples of parts of planes.The shelves are the same distance apart at all points, and do not appear tointersect.They are _______.parallelIn geometry, planes that do not intersect are called _____________.parallel planesPlane PSR || plane JMLPlane JPQ || plane MLRPlane PJM || plane QRL

Parallel Lines and PlanesSometimes lines that do not intersect are not in the same plane.These lines are called __________.skew lines

Definition ofSkewLinesTwo lines that are not in the same plane are skew iff they do not intersect.

Parallel Lines and PlanesName the parts of the figure:1) All planes parallel to plane ABFPlane DCG

Parallel Lines and TransversalsYou will learn to identify the relationships among pairs of interior and exterior angles formed by two parallel linesand a transversal.

Parallel Lines and TransversalsIn geometry, a line, line segment, or ray that intersects two or more lines atdifferent points is called a __________transversalNote all of the different angles formed at the points of intersection. 12345768

Parallel Lines and TransversalsThe lines cut by a transversal may or may not be parallel.

Definition ofTransversalIn a plane, a line is a transversal iff it intersects two or moreLines, each at a different point.

Parallel Lines and TransversalsTwo lines divide the plane into three regions. The region between the lines is referred to as the interior.The two regions not between the lines is referred to as the exterior.

Parallel Lines and TransversalsWhen a transversal intersects two lines, _____ angles are formed.eightThese angles are given special names.tInterior angles lie between thetwo lines.Exterior angles lie outside thetwo lines.Alternate Interior angles are on the opposite sides of the transversal.Consectutive Interior angles are on the same side of the transversal.Alternate Exterior angles areon the opposite sides of thetransversal.

Parallel Lines and Transversals12345768congruent

Theorem 4-1AlternateInteriorAnglesIf two parallel lines are cut by a transversal, then each pair ofAlternate interior angles is _________.

Parallel Lines and Transversalssupplementary

Theorem 4-2ConsecutiveInteriorAnglesIf two parallel lines are cut by a transversal, then each pair ofconsecutive interior angles is _____________.

Parallel Lines and Transversalscongruent

Theorem 4-3AlternateExteriorAnglesIf two parallel lines are cut by a transversal, then each pair ofalternate exterior angles is _________.

Practice Problems:

1, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, 38, 40, 42, 44, and 46 (total = 23)

Transversals and Corresponding AnglesYou will learn to identify the relationships among pairs of corresponding angles formed by two parallel lines and atransversal.

Transversals and Corresponding AnglesWhen a transversal crosses two lines, the intersection creates a number ofangles that are related to each other.Note 1 and 5 below. Although one is an exterior angle and the other is an interior angle, both lie on the same side of the transversal. Angle 1 and 5 are called __________________.corresponding anglesGive three other pairs of corresponding angles that are formed:4 and 83 and 72 and 6

Transversals and Corresponding Anglescongruent

Postulate 4-1CorrespondingAnglesIf two parallel lines are cut by a transversal, then each pair ofcorresponding angles is _________.

Transversals and Corresponding Anglesalternate interioralternate exteriorcorrespondingconsecutive interiorTypes of angle pairs formed when a transversal cuts two parallel lines.

ConceptSummaryCongruentSupplementary

Transversals and Corresponding Angless || t and c || d.

Name all the angles that arecongruent to 1.Give a reason for each answer.3 1corresponding angles6 1vertical angles8 1alternate exterior angles 9 1corresponding angles11 9 1corresponding angles14 1alternate exterior angles16 14 1corresponding angles

Practice Problems: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26, 28, 30, 32, 34, 36, and 38 (total = 19)

Proving Lines ParallelYou will learn to identify conditions that produce parallel lines.Reminder: In Chapter 1, we discussed if-then statements (pg. 24).Within those statements, we identified the __________ and the _________.hypothesisconclusionI said then that in mathematics, we only use the term if and only ifif the converse of the statement is true.

Proving Lines ParallelPostulate 4 1 (pg. 156):IF ___________________________________,

THEN ________________________________________.two parallel lines are cut by a transversaleach pair of corresponding angles is congruentThe postulates used in 4 - 4 are the converse of postulates that you alreadyknow. COOL, HUH?4 4, Postulate 4 2 (pg. 162):IF ________________________________________,

THEN ____________________________________.each pair of corresponding angles is congruenttwo parallel lines are cut by a transversal

Proving Lines ParallelparallelIf 1 2,

then _____a || b

Postulate 4-2In a plane, if two lines are cut by a transversal so that a pairof corresponding angles is congruent, then the lines are _______.

Proving Lines ParallelparallelIf 1 2,

then _____a || b

Theorem 4-5In a plane, if two lines are cut by a transversal so that a pairof alternate interior angles is congruent, then the two lines are _______.

Proving Lines ParallelparallelIf 1 2,

then _____a || b

Theorem 4-6In a plane, if two lines are cut by a transversal so that a pairof alternate exterior angles is congruent, then the two lines are _______.

Proving Lines ParallelparallelIf 1 + 2 = 180,

then _____a || b

Theorem 4-7In a plane, if two lines are cut by a transversal so that a pairof consecutive interior angles is supplementary, then the two lines are _______.

Proving Lines ParallelparallelIf a t and b t,

then _____a || b

Theorem 4-8In a plane, if two lines are cut by a transversal so that a pairof consecutive interior angles is supplementary, then the two lines are _______.