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Transcript of Geometry
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GeometryGeometry
3.3 Proving Lines Parallel3.3 Proving Lines Parallel
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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~
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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 //~
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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 ~
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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 ~
//
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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
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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 //
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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.
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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
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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.
.
.
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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
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In Summary (the key ideas)………In Summary (the key ideas)………
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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
~
~
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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
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HomeworkHomework
pg. 87 # 1-27 oddpg. 87 # 1-27 odd