Transcript Intro to Proofs - CrockettGeometryStudent

Intro to Mathematical Proofs
With the help of some awesome
plagiarism!
Parallel lines
• Same ________
• Different ______
• Parallel lines will
never___________
Parallel lines
• Same slope
• Different y-intercept
• Parallel lines will
never intersect.
• If lines l1 and l2 are
parallel we say l1 l2
Perpendicular lines
• Have _______
________ slopes
• Will intersect at
______ points, and
form a ______
angle.
Perpendicular lines
• Have opposite
reciprocal slopes
• Will intersect at
exactly 1 point, and
form a 90〫angle.
• If lines l1 and l2
are perpendicular
we say l1l2
Perpendicular or Parallel?
y
3
x3
2
y  3020x  1203



2
y  x3
3
and
y  3020x

and
y  90210
30
y   x 15
2
and
y 15x 
15
x
11
and
y
y  2x  41
and
y  8,675,309
y 

and



30
2
11
x 1337
15
y  41 2x
How’d you do?
3
x3
2
and
2
y  x3
3
Perp
and
y  3020x
Parallel
and
y  90210
Parallel
30
y   x 15
2
and
y 15x 
15
x
11
and
y
y  2x  41
and
y
y  3020x  1203




y  8,675,309
y 




30
2
Neither
11
x 1337 Perp
15
y  41 2x
Neither
Parallel lines with a perpendicular transversal
l k and tk so tl
How would we say that?


Transversal means:

Parallel lines with a perpendicular transversal
lk


and
tk so tl
This is a short proof which takes
advantage of the theorem: “if a
transversal intersects two parallel
lines, then each pair of alternate
angles are equal.”

Don’t worry if that didn’t make
sense! Just take away that there
is a rule that tells us if lines l and
k are parallel, and t is
perpendicular to k, then t is
also perpendicular to l.
Geometry
Geometry is a refined
system built on a few rules
such as, “Given any two
distinct points, there is
exactly one line that
connects them.” From these
basic rules we find many
consequences.
Euclid: Mac-Daddy of Math
Geometry
While geometric reasoning had
been used for hundreds of years
before him, Euclid is considered
the Father of modern geometry.
In 300 BC this dude wrote
Elements. These books did not
only mean “This is how
Geometry will be” but ,“this is
how all mathematics will be set
up forever. The end. Go home.”
Geometry
In book one of Elements Euclid used ten postulates
(rules) and from them proved many theorems, such as
the Pythagorean Theorem; the Vertical Angle Theorem;
and the interior-angle sum of a triangle is 180 〫.
While the Pythagorean Theorem had been used for
thousands of years before Euclid, Euclid was the first to
systematically prove every theorem which the
Pythagorean Theorem relied on.
Vertical Angle Theorem
The Vertical Angle Theorem
states: the angles formed by
two transversal lines will form
vertical congruent pairs.
“∠ 1” means angle 1
For example: ∠ 1 and ∠ 3 will
have the same measure. As
will ∠ 2 and it’s vertical
partner, which is 120〫.
Vertical Angle Theorem
The Vertical Angle Theorem states:
the angles formed by two transversal
lines will form vertical congruent
pairs.
Now we know:
∠ 2 = 120 〫
∠ 1 ∠ 3 (read ∠1 is congruent to
∠3)
We also know every straight
line has an angle of _____〫
Vertical Angle Theorem
The Vertical Angle Theorem states:
the angles formed by two transversal
lines will form vertical congruent
pairs.
We also know every straight line has
an angle of 180〫
Now we know:
∠ 2 = 120 〫
∠1 ∠ 3
∠ 3 + 120 〫= 180 〫
Vertical Angle Theorem
Now we know:
∠ 2 = 120 〫
∠ 1 ∠ 3
∠ 3 + 120 〫 = 180 〫
So,
∠ 3 = 180 〫- 120 〫 = __
〫
And
∠ 1 = __ 〫
Vertical Angle Theorem
Now we know:
∠ 2 = 120 〫
∠ 1 ∠ 3
∠ 3 + 120 〫 = 180 〫
So,
∠ 3 = 180 〫 - 120 〫 =
60〫
And
∠ 1 = 60〫
Alternate-Interior-Angle Theorem
Let’s start out with what we
already know about these
angles from the Vertical Angle
Theorem:
∠ a …
Alternate-Interior-Angle Theorem
The actual proof for the
Alternate-Interior-Angle
Theorem requires the use of
contra-positive reasoning.
Basically, this means we show
the theorem is true by lying
first. We say the theorem is
false, then reach an
impossible conclusion as a
result!
Alternate-Interior-Angle Theorem
For the sake of time we are
going to assume the AlternateInterior-Angle Theorem is true!
The Alternate-Interior-Angle
Theorem (AIA) tells us if a
transversal line crosses two
parallel lines then the alternate
interior angles are congruent.
Alternate-Interior-Angle Theorem
The Alternate-Interior-Angle
Theorem (AIA) tells us if a
transversal line crosses two
parallel lines then the alternate
interior angles are congruent.
So:
∠d ∠e
Now use the AIA and vertical
angles to show ∠a ≅ ∠h ∠
Alternate-Interior-Angle
Theorem
Now show ∠d + ∠f = 180˚
Use this conclusion to show
every parallelogram has an
interior angle sum of 360˚
Proving the interior angle sum
of a triangle is 180 〫
Now using:
Alternate-Interior-Angle Theorem;
a
Vertical Angle Theorem;
And the fact that straight lines have
b
c
an angle measure of 180 〫
Show that the interior
angle sum of every
triangle is 180 〫
Proving the interior angle sum
of a triangle is 180 〫
a
b
c
Proving the interior angle sum
of a triangle is 180 〫
b
b
Proving the interior angle sum
of a triangle is 180 〫
b
a
b
c
Proving the interior angle sum
of a triangle is 180 〫
c
c
Proving the interior angle sum
of a triangle is 180 〫
b
c
a
b
c
Proving the interior angle sum
of a triangle is 180 〫
b
c
a