# Theorems

## The Distance Formula $d=\sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Solution: Let A be point 1 and B be point 2. We have $x_1\ =5,y_1 = -7, x_2 = -6, y_2 = -2$

Substituting we have $d=\sqrt{(\ -6 -5)^2 + (-2 - -7)^2} = \sqrt{(\ -11)^2 + (-2 +7)^2} = \sqrt{(\ -11)^2 + (+5)^2} = \sqrt{\ 121 + 25} = \sqrt{146}$

Try some distance problems at Distance & Midpoint 9-2 ME Worksheet

## Sum of Two Sides of a Triangle

### Thm: The sum of two sides of any triangle must be greater than the third side

Example: Given two sides of a triangle are 8 inches and 11 inches, find all the possible lengths of the third side (also known as S3)

Solution: Since we know that 8 + 11 must be greater than S3, we have $8\ + 11 > S_3$, so S3 must be less than 19. But what is the smallest value that S3 can be?

Well, we know that $S_3\ + 11\ must\ be > 8$, but 11 is already greater than 8 so that doesn't help us. However we also know that $S_3\ + 8\ must\ be > 11$.

Using algebra we subtract 8 from both sides of our inequality we get $S_3\ + 8 {\color{Red}-8}\ > 11 {\color{Red}-8}$, so $S_3\ > 3$. Our final answer is $3 < S_3\ < 19$ (our third side must be greater than 3 but less than 19).

## External Angle Theorem

### Thm: An external angle is equal to the sum of the two opposite interior angles.

Look at the figure below. We know that the sum of the interior angles of a triangle is 180 degrees. That means that in triangle ABC below, $\angle 2 + \angle 3 + \angle 4 = 180^\circ$. We also know that angles 1 and 2 are supplementary angles, meaning that $\angle 2 + \angle 1 = 180^\circ$. So if $\angle 2 + \angle 1 = 180^\circ$ and $\angle 2 + \angle 3 + \angle 4 = 180^\circ$, then it must be true that $\angle 1 = (\angle 3 + \angle 4)$ by substitution.

## Vertical Angle Theorem

### Vertical angles are equal

Look at the intersecting line segments EF & GH below. We know that $\angle 1 + \angle 2 = 180^\circ$ since they're supplementary angles. We also know that $\angle 1 + \angle 4 = 180^\circ$ because they too are supplementary. If that's the case, it must be true that $\angle 2 = \angle 4$ by substitution. By the same logic, $\angle 1 = \angle 3$. Angle 2 and angle 4 are vertical angles, and angle 1 and angle 3 are also vertical angles.

## SAS Triangle Congruence Theorems

### If 2 sides and the included angle of one triangle are equal in length to 2 sides and the included angle of another triangle, the two triangles are congruent. This is called the Side-Angle-Side triangle congruence theorem, or the SAS theorem.

Important note: The included angle is the angle between the two corresponding sides.

## The Sum of the Interior Angles of any Polygon

To find the sum of the interior angles of any polygon we use the following formula: $(n-2)180^\circ$

Example: Find the sum of the interior angles of a hexagon (six-sided figure)

Solution: Since a hexagon has 6 sides, we substitute n with 6 which gives us $(6-2)180 = (4)180 = 720^\circ$ so the sum of the interior angles of a hexagon is $720^\circ$

## The Measure of Each Interior Angle of a Regular Polygon

Since a regular polygon has sides of equal length and angles of equal measure (all sides are equal and all angles are equal), we divide the sum of the interior angles by the number of angles.

Since the number of angles and the number of sides is the same, we use the following formula to find the measure of each angle of the regular polygon: $\frac{(n-2)180}{n}$

Example: Find the number of degrees in each interior angle of a regular octagon.

Solution: Since an octagon has 8 sides, we substitute n with 8. $\frac{(8-2)180}{8} = \frac{(6)180}{8} = \frac{1080}{8} = 135^\circ$

So the measure of each interior angle of a regular octagon is 135 degrees.

## The Measure of Each Exterior Angle of a Regular Polygon

There are two ways to find the measure of an exterior angle of a regular polygon.

Since the sum of each interior and exterior angle of a regular polygon is 180 degrees, we can simply calculate the measure of an interior angle of a regular polygon and subtract that angle from 180 degrees to find the measure of the exterior angle.

Example: Find the measure of each exterior angle of a regular dodecagon (12 sided figure).

Solution 1: We can calculate the measure of each interior angle of a regular dodecagon by using the formula $\frac{(n-2)180}{n}$. By substitution we get $\frac{(12-2)180}{12} = \frac{(10)180}{12} = \frac{1800}{12} = 150^\circ$. We can now subtract 150 degrees from 180 degrees and our answer is $30^\circ$.

Solution 2: Since the sum of all the exterior angles of any polygon is 360 degrees, we can simply divide 360 by the number of sides in our polygon. We know that a dodecagon has 12 sides so $\frac{360}{12}=30^\circ$