parallel lines advanced geometry parallel and perpendicular lines lesson 1

Parallel Lines

Advanced GeometryParallel and Perpendicular Lines

Lesson 1

Lines (segments & rays, also) are either

intersecting,

or skew.

l m

parallel,

AB CD

and

are skew.

do not intersect

Skew Lines

are not coplanar

Planes can be parallel

OR

intersecting.

Name all segments that are parallel to .

Name all segments that are skew to .

Name all segments that intersect .

AF

GH

FG

Example:

Name all planes that are parallel to plane DEF.

, ,DE BG CH

, , ,BG FG CH EH

, , , ,AD DE BC CH and CE

Figure from p. 143.

TransversalIdentify the pairs of lines to which each

given line is a transversal.

m

p

l and n; l and p; n and p

l and m; l and n; m and n

Whenever there are two lines and a transversal, eight angles are formed.

interior angles – angles between the two lines

exterior angles – angles outside the two lines

∠3, ∠4, ∠5, ∠6 ∠1, ∠2, ∠7, ∠8

a

b

t

Alternate Interior Angles

•different sides of the transversal

AIA

•both angles are interior

•different sides of the transversal

AlternateExterior Angles AEA

•both angles are exterior

∠3 and ∠6∠4 and ∠5

∠1 and ∠8∠2 and ∠7

a

b

t

Alternate Interior Angles

Alternate Interior Angles

AlternateExterior AnglesAlternateExterior Angles

Consecutive Interior Angles

•same side of the transversal

CIA

•both angles are interior

•same side of the transversal

Corresponding Angles

•one interior angle and one exterior angle

∠3 and ∠5∠4 and ∠6

∠1 and ∠5∠2 and ∠6∠3 and ∠7∠4 and ∠8

a

b

t

Consecutive Interior Angles

Consecutive Interior Angles

Identify each pair of angles and AIA, AEA, CIA, corresponding angles, vertical angles, or a linear pair.

∠5 and ∠12

∠10 and ∠3

∠4 and ∠7

∠10 and ∠11

∠2 and ∠12

∠3 and ∠14

AEA

corresponding angles

vertical angles

CIA

linear pair

AIA

Name the transversal that forms each pair of angles. Then identify the pair of angles.

∠8 and ∠6

∠10 and ∠14

∠2 and ∠16

∠10 and ∠11

corresponding angles

AIA

AEA

CIA

a

b

c

b

When the lines cut by a transversal are parallel,the four angle pairs have a special relationship.

•AIA are congruent.

•AEA are congruent.

•CIA are supplementary.

•Corresponding angles are congruent.

Reminder:•Vertical angles are ALWAYScongruent.•Linear pairs are ALWAYS supplementary.

In the figure a || b and m 4 = 51. Find the measures ∠of all numbered angles. Justify each answer.

a

b

Find the measure of each angle if m∠1 = 62 and m∠7 = 107.

What is the measure of ∠RTV?

If m∠2 = 4x + 7 and m 3 = 5x – 13, find ∠ m 3.∠

If m∠5 = 2x – 10, m 6 = 4(y – 25), ∠and m 7 = x + 15. find x and y.∠