Theorems
From MHSHS Wiki
(→Thm: An external angle is equal to the sum of the two opposite interior angles.) |
(→Thm: An external angle is equal to the sum of the two opposite interior angles.) |
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Look at the figure below. We know that the sum of the interior angles of a triangle is 180 degrees. That means that | Look at the figure below. We know that the sum of the interior angles of a triangle is 180 degrees. That means that | ||
- | <math>\angle 2 + \angle 3 + \angle 4 = 180^\circ</math>. We also know that angles 1 and 2 are supplementary angles, meaning that <math>\angle 2 + \angle 1 = 180^\circ</math>. | + | in triangle ABC below, <math>\angle 2 + \angle 3 + \angle 4 = 180^\circ</math>. We also know that angles 1 and 2 are supplementary angles, meaning that <math>\angle 2 + \angle 1 = 180^\circ</math>. |
So if <math>\angle 2 + \angle 1 = 180^\circ</math> and <math>\angle 2 + \angle 3 + \angle 4 = 180^\circ</math>, then it must be true that <math>\angle 1 = (\angle 3 + \angle 4)</math> by substitution. | So if <math>\angle 2 + \angle 1 = 180^\circ</math> and <math>\angle 2 + \angle 3 + \angle 4 = 180^\circ</math>, then it must be true that <math>\angle 1 = (\angle 3 + \angle 4)</math> by substitution. | ||
Revision as of 00:51, 21 October 2009
The Distance Formula
Example: Find the distance between point A(5,-7) and B(-6,-2) and leave your answer in radical form.
Solution: Let A be point 1 and B be point 2. We have
Substituting we have
Try some distance problems at Distance & Midpoint 9-2 ME Worksheet
Sum of the 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 , so S3 must be less than 19. But what is the smallest value that S3 can be?
Well, we know that , but 11 is already greater than 8 so that doesn't help us.
However we also know that .
Using algebra we subtract 8 from both sides of our inequality we get , so . Our final answer is (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, . We also know that angles 1 and 2 are supplementary angles, meaning that . So if and , then it must be true that by substitution.