geometry

GeometryGeometry

3.3 Proving Lines Parallel3.3 Proving Lines Parallel

PostulatePostulateFrom From yesterdayyesterday : :

If two // lines are cut by a transversal, If two // lines are cut by a transversal, then corresponding angles are then corresponding angles are

congruent.congruent.

// Lines => corr. <‘s = ~

1 2

3 4

5 6

7 8

<1 = <5~

PostulatePostulateTodayToday, we learn its , we learn its converseconverse : :

If two lines are cut by a transversal If two lines are cut by a transversal and and corresponding angles are congruent, corresponding angles are congruent,

then the lines are parallel.then the lines are parallel.

corr. <‘s = => // Lines~1 2

3 4

5 6

7 8

If <1 = <5, then lines are //~

TheoremTheoremFrom From yesterdayyesterday::

If two // lines are cut by a transversal, then If two // lines are cut by a transversal, then alternate interior angles are congruent.alternate interior angles are congruent.

// Lines => alt int <‘s = ~

1 2

3 4

5 6

7 8

Example: <3 = <6 ~

TheoremTheoremTodayToday, we learn its , we learn its converseconverse : :

If two lines are cut by a transversal and If two lines are cut by a transversal and alternate interior angles are congruent, alternate interior angles are congruent, then the lines are parallel.then the lines are parallel.alt int <‘s = => // Lines~

1 2

3 4

5 6

7 8

If <3 = <6, then lines are ~

//

TheoremTheorem

From From yesterdayyesterday::

If two // lines are cut by a transversal, then same If two // lines are cut by a transversal, then same side interior angles are supplementary.side interior angles are supplementary.

// Lines => SS Int <‘s supp

1 2

3 4

5 6

7 8

Example: <4 is supp to <6

TheoremTheoremTodayToday, we learn its , we learn its converseconverse : :

If two lines are cut by a transversal and same If two lines are cut by a transversal and same side interior angles are supplementary, then the side interior angles are supplementary, then the lines are parallel .lines are parallel .

SS Int <‘s supp => // Lines1 2

3 4

5 6

7 8

If <4 is supp to <6, then the lines are //

TheoremTheoremFrom From yesterdayyesterday::

If a transversal is perpendicular to one of If a transversal is perpendicular to one of two parallel lines, then it is perpendicular two parallel lines, then it is perpendicular to the other line.to the other line.

TheoremTheoremTodayToday, we learn its , we learn its converseconverse: :

In a plane two lines perpendicular to the In a plane two lines perpendicular to the same line are parallel.same line are parallel.

If k and l are both to t then the lines are //

k

l

t

3 More Quick TheoremsTheorem: Through a point outside a line,

there is exactly one line parallel to the given line.

Theorem: Through a point outside a line, there is exactly one line perpendicular to the given line.

Theorem: Two lines parallel to a third line are parallel to each other.

.

.

Which segments are parallel ?…

W H A T

L I N E

23

61

22

62

Are WI and AN parallel?

No, because <WIL and <ANI are not congruent

61 ≠ 62

Are HI and TN parallel?

Yes, because <WIL and <ANI are congruent

61 + 23 = 8462 + 22 = 84

In Summary (the key ideas)………In Summary (the key ideas)………

5 Ways to Prove 2 Lines Parallel5 Ways to Prove 2 Lines Parallel

1.1. Show that a pair of Show that a pair of Corr. <‘s are =Corr. <‘s are =

2.2. √ √ √ √ √ √ √ √ √ √ Alt. Int. <‘s are =Alt. Int. <‘s are =

3.3. √ √ √√ √√ √√ √√ S-S Int. <‘s are suppS-S Int. <‘s are supp

4.4. Show that 2 lines are Show that 2 lines are to a 3to a 3rdrd line line

5.5. √ √ √ √ √ √ √ √ √ √ to a 3to a 3rdrd line line

~

~

Turn to pg. 87Turn to pg. 87

Let’s do #19 and # 28 from your homework Let’s do #19 and # 28 from your homework togethertogether

HomeworkHomework

pg. 87 # 1-27 oddpg. 87 # 1-27 odd