# Theorems, Corollaries, Lemmas

What are all those things? They sound so impressive!

Well, they are basically just **facts**: some result that has been arrived at.

- A Theorem is a
**major**result - A Corollary is a theorem that
**follows on**from another theorem - A Lemma is a
**small**result (less important than a theorem)

## Examples

Here is an example from Geometry:

### Example: A Theorem and a Corollary

#### Theorem:

Angles on one side of a straight line always add to 180°.

#### Corollary:

Following on from that theorem we find that where two lines intersect, the angles opposite each other (called Vertical Angles) are **equal** (a=c and b=d in the diagram).

Angle a = angle c

Angle b = angle d

#### Proof:

Angles a and b add to 180° because they are along a line:

a + b = 180°

a = 180° − b

Likewise for angles b and c

b + c = 180°

c = 180° − b

And since both a and c equal 180° − b, then

a = c

And a slightly more complicated example from Geometry:

### Example: A Theorem, a Corollary to it, and also a Lemma!

#### Theorem:

An inscribed angle a° is half of the central angle 2a°

Called the **Angle at the Center Theorem**.

**Proof: Join the center O to A.**

Triangle ABO is isosceles (two equal sides, two equal angles), so:

**b°**

And, using Angles of a Triangle add to 180°:

Triangle ACO is isosceles, so:

**c°**

And, using Angles of a Triangle add to 180°:

And, using Angles around a point add to 360°:

Replace **b + c** with **a**, we get:

Angle BAC = a° and Angle BOC = 2a°

And we have proved the theorem.

(That was a "major" result, so is a Theorem.)

#### Corollary

(This is called the *"Angles Subtended by the Same Arc Theorem*", but it’s really just a **Corollary** of the *"Angle at the Center Theorem"*)

Keeping the endpoints fixed ... ... the angle a° is always the same, no matter where it is on the circumference:

So, Angles Subtended by the Same Arc are equal.

#### Lemma

(This is sometimes called the *"Angle in the Semicircle Theorem"*, but it’s really just a **Lemma** to the *"Angle at the Center Theorem"*)

In the special case where the central angle forms a diameter of the circle:

2a° = 180° , so a° = 90°

So an angle inscribed in a semicircle is always a right angle.

(That was a "small" result, so it is a Lemma.)

Another example, related to Pythagoras' Theorem:

### Example:

#### Theorem

If m and n are any two whole numbers and

- a = m
^{2}− n^{2} - b = 2mn
- c = m
^{2}+ n^{2}

then a^{2} + b^{2} = c^{2}

**Proof**:

**a**= (m

^{2}+ b^{2}^{2}− n

^{2})

^{2}+ (2mn)

^{2}

^{4}− 2m

^{2}n

^{2}+ n

^{4}+ 4m

^{2}n

^{2}

^{4}+ 2m

^{2}n

^{2}+ n

^{4}

^{2}+ n

^{2})

^{2}

**= c**

^{2}(That was a "major" result.)

#### Corollary

a, b and c, as defined above, are a Pythagorean Triple

**Proof**:

From the Theorem **a ^{2} + b^{2} = c^{2}**,

so a, b and c are a Pythagorean Triple

(That result "followed on" from the previous Theorem.)

#### Lemma

If m = 2 and n = 1, then we get the Pythagorean triple 3, 4 and 5

**Proof**:

If m = 2 and n = 1, then

- a = 2
^{2}− 1^{2}= 4 − 1 =**3** - b = 2 × 2 × 1 =
**4** - c = 2
^{2}+ 1^{2}= 4 + 1 =**5**

(That was a "small" result.)