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 elope.

• If lines and are parallel we say

€

l1

€

l2

€

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 90 〫 angle.

• If lines and are perpendicular we say

€

l1

€

l2

€

l1⊥l2

Perpendicular or Parallel?

and

and

and

and

and

and

€

y =3

2x + 3

€

y = −2

3x + 3

€

y = 3020x +1203

€

y = 3020x

€

y = 8,675,309

€

y = 90210

€

y = −30

2x −15

€

y =15x +30

2

€

y = −15

11x

€

y =11

15x +1337

€

y = −2x + 41

€

y = 41+ 2x

How’d you do?

and

and

and

and

and

and

€

y =3

2x + 3

€

y = −2

3x + 3

€

y = 3020x +1203

€

y = 3020x

€

y = 8,675,309

€

y = 90210

€

y = −30

2x −15

€

y =15x +30

2

€

y = −15

11x

€

y =11

15x +1337

€

y = −2x + 41

€

y = 41+ 2x

Perp

Parallel

Parallel

Neither

Perp

Neither

Parallel lines with a perpendicular transversal

and so

How would we say that?

Transversal means:€

l k

€

t⊥k

€

t⊥l

Parallel lines with a perpendicular transversal

and so

€

l k

€

t⊥k

€

t⊥lThis 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.

GeometryGeometry 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

GeometryWhile 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.”

GeometryIn 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 120 〫 .

Vertical Angle TheoremThe 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 TheoremThe 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 180 〫

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 〫〫 = 60 〫And⦟1 = 60 〫

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 Alternate-Interior-Angle Theorem is true!

The corresponding angle theorem tells us if a transversal line crosses two parallel lines then a set of corresponding congruent angles are formed.

Alternate-Interior-Angle Theorem

The Alternate-Interior-Angle Theorem tells us if a transversal line crosses two parallel lines then a set of corresponding congruent angles are formed.

So:⦟d ≅ e⦟and… from there we can use the Vertical Angle Theorem to show:⦟a ≅ e⦟⦟g ≅ c⦟

Proving the interior angle sum of a triangle is 180 〫

Now using:

Alternate-Interior-Angle Theorem;

Vertical Angle Theorem;

And the fact that straight lines have

an angle measure of 180 〫

Show that the interior angle sum of a triangle is 180 〫〫