This is illustrated in the image below:

We see two parallel lines and a third line (transversal) intersecting (crossing or cutting through) both of them. The green shaded angles are: (1) inside (between) the two parallel lines, (2) congruent (identical or the same), and (3) on opposite sides of the transversal.

This is true for the other two unshaded interior angles. It is also true for the alternate exterior angles (but not proved here).

Axioms

Proofs are built on two things: (1) postulates, axioms, or hypotheses – these are things that are assumed to be true, but can’t be proved. In general, we try to use the fewest number of axioms we can; (2) other proofs.

This proof depends on two axioms: (1) if you pick any two distinct points on a straight line, the angle between those two points will be 180°; (2) if you take any two intersecting straight lines and shift one of the lines so it it is in a different position, but still parallel to its original position, the angle between the two intersecting lines stays the same.

Proof

1) Consider two intersecting lines:

At the point of intersection, there is an angle (which I call A).

2) If the line is straight, the angle between any two points on that line must be 180° (axiom #1 from above):

3) If one angle is A degrees, then obviously the other angle must be 180° – A:

4) This is true for both straight lines:

5) This means the remaining angle must be 180° – (180° – A) = A degrees:

This proves that angles on alternate sides of the transversal (at the point of intersection) are congruent (identical).

6) If we draw a line parallel to one of the lines (it doesn’t matter which) and it intersects the other line (that line is now called a transversal), we know that the angles of intersection must be the same (axiom #2 from above).

If the angles of intersection are the same for both lines, then the alternate interior angles must be the same:

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Proving that the angles inside a triangle – any triangle – sum up to 180° is very simple, but leaves most people unsatisfied (or unconvinced) because it depends on the properties of something called Alternate Interior Angles.

I’ll give the proof first and then explain Alternate Interior Angles

A demonstration of the angles of a triangle summing up to 180° can be found here.

Proof

Given any triangle, how can you prove that the angles inside a triangle sum up to 180°?

1) Draw a line parallel to one of the sides of the triangle that passes through the corner opposite to that side:

2) This creates two new angles. We’ll call them D and E:

3) We know by Alternate Interior Angles that angle D must be the same as angle A and angle E must be the same as angle C.

4) Replace the new angles found in step 2) with A and C:

5) We see that angle A + angle B + angle C add up to give us a straight line or 180°:

Alternate Interior Angles

When two lines intersect (cross each other), you get four angles: A, B, C, and D:

It happens that angles A and C are identical and angles B and D are identical.

If we move one of the lines so it remains parallel to its original position (in math-speak, this is called translation), what happens to the angles? Nothing. The angles remain the same. It doesn’t matter where the two lines intersect (cross). Only the angle of intersection matters.

We don’t have to move the line, we can just draw a line parallel to it – we can draw as many parallel lines as we like (but that is likely to confuse things). Each intersection (crossing) has the same angles:

We saw earlier, that A and C are the same angle (as well, as B and D):

We call the angles between the parallel lines interior angles because they are inside (in between) the parallel lines.

We notice that the angles on opposite sides of the transversal (math-speak for the line that intersects (crosses) the parallel lines) are the same. We say the angles on alternate sides of the line are identical.

Put his all together and you get Alternate Interior Angle

Knowing all this, when we take another look at the proof that all the angles in a triangle sum to 180° we see the following:

1) the base of the triangle forms one of the parallel lines and we draw the second.

2) each of the sides of the triangle acts like a transversal intersecting the parallel lines.

Because angles on alternate sides of the transversal are identical, we are able to sum all three angles and show that they sum up to 180°

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We learned the Pythagorean Theorem in grade school: The square of the length of the hypotenuse is equal to the sum of the squares of the other two sides. Often written as

A² + B²= C².

The Pythagorean Theorem is much more than just a mathematical curiosity, it is a basic and useful mathematical property that shows up as the solution to many mathematical problems or in their proofs (the other two big ones are the right angle triangle and the trigonometric functions – but they are all related).

There are over 300 different proofs for the Pythagorean Theorem, I like this one best. It is simple and only requires some basic knowledge about triangles and some elementary algebra.

1a) Begin with a right angle triangle:

1b) Since this is a right angle triangle, we know one of the angles is 90 degrees (coloured in blue)

1c) For the second angle, we can choose any angle we want. We will call it α (coloured in green)

1d) Because this is a triangle, we know all the angles must add up to 180 degrees. We can use this to find the third angle (coloured in red):

• 90 + α + unknown = 180
• unknown = 180 – 90 – α (grouping the known terms and unknown terms together)
• unknown = 90 – α (simplifying)

2) Rotate the triangle so the hypotenuse is horizontal (lying flat). This step isn’t necessary, but makes it easier to see what is happening.

3a) Draw a perpendicular line from the hypotenuse through the corner opposite it. With the triangle rotated, it is simply a matter of dropping a vertical line from the peak to the hypotenuse.

3b) This perpendicular line (which I call p) divides the original triangle into two smaller triangles. Side p becomes one of the sides of both smaller triangles.

3c) The whole of side A went to the triangle on the left hand side.

3d) The whole of side B went to the triangle on the right hand side.

3e) Side C was divided in two. We’ll call the part that went to the left hand side triangle x and the part that went to the right hand side triangle y. Remember that the length of side x plus the length of side y is equal to the length of side C. Mathematically we would write:

x + y = C

The left hand side (LHS) triangle:

4a) We know that one of the angles is 90 degrees. Because line p is perpendicular to the hypotenuse, the angle must be 90 degrees (coloured blue).

4b) When we divided the original triangle, the angle α remained in the LHS triangle, so we know this angle is α (coloured green).

4c) Since the angles in a triangle must add up to 180 degrees, we know that the remaining angle must be 90 – α degrees (coloured red).

4d) Since both the original triangle and the LHS triangle have the same angles, both triangles are similar triangles.

The right hand side (RHS) triangle:

4e) We know one of the angles is 90 degrees because line p is perpendicular to the hypotenuse (coloured blue).

4f) When we divided the triangle, the angle 90 – α remained in the RHS triangle, so we know this angle is 90 – α degrees (coloured red).

4g) Since the angles in a triangle must add up to 180 degrees, we know that the remaining angle must be α degrees (coloured green).

4h) Since both the original triangle and the RHS triangles have the same angles, they must be similar triangles.

5) In dividing the original triangle, we have created two smaller similar triangles (the original triangle has been rotated and flipped, the LHS triangle has been rotated – I left the text as is):

6) Another property of similar triangles is that the ratio of the lengths of the same sides is the same:

• the ratio of A to B must be the same as
• the ratio of x to p which must be the same as
• the ratio of p to y

Mathematically:

A/B = x/p = p/y

7) We know that the ratio A/C (of the original triangle) = x/A (of the LHS triangle): A/C = x/A.

By cross multiplying, we get: A² = Cx

8) We know that the ratio B/C (of the original triangle) = y/B (of the RHS triangle): B/C = y/B

By cross multiplying, we get B² = Cy

Where the magic happens

Fundamental to basic algebra is the ability to manipulate an equation any way we like – as long as we do the same thing on both sides.

For example, consider 4 = 4:

We can add 8 to both sides and the relationship remains the same: 4 + 8 = 4 + 8.

We can do the same with our equation from step (7): A² = Cx

If we add 33 to both sides we get: A² + 33 = Cx + 33, but we don’t want to do that because it doesn’t bring us any closer to proving the Pythagorean Theorem

What we want to do, is somehow bring B into the equation (we already have A and C).

We can try A² + B² = Cx + B², but that doesn’t help much (the B² cancels out).

We need to employ an algebraic trick – we don’t want to add the same thing to both sides of the equation, we want to add equivalent things.

For example, consider 4 = 4. If we add 8 to both sides we would typically do it this way: 4 + 8 = 4 + 8. However, it is also perfectly valid to do: 4 + 8 = 4 + 1 + 2 + 5 (this is one of those non-intuitive mathematical tricks – but it is very useful).

Let us add B² to the equation from step (7), but this time, we will add the equivalent of B² to the right hand side. From step (8) we know that B² = Cy

Using the trick of adding equivalent values to both sides of the equation we get:

A² + B² = Cx + Cy

This is very good. We can simplify the right hand side by extracting a common factor:

A² + B² = C(x + y)

When we divided the original triangle in step (3a), we divided line C into two parts: x and y. From (3e) we know that x + y = C, so we can replace x + y in the equation above by C:

A² + B² = C(C)

When simplified gives us:

A² + B² = C²

which is the Pythagorean Theorem.

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