Theorems
From MHSHS Wiki
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==Vertical Angle Theorem== | ==Vertical Angle Theorem== | ||
+ | ===Vertical angles are equal=== | ||
Look at the intersecting line segments EF & GH below. We know that <math>\angle 1 + \angle 2 = 180^\circ</math> since they're supplementary angles. We also know that <math>\angle 1 + \angle 4 = 180^\circ</math> because they too are supplementary. If that's the case, it must be true that <math>\angle 2 = \angle 4</math> by substitution. | Look at the intersecting line segments EF & GH below. We know that <math>\angle 1 + \angle 2 = 180^\circ</math> since they're supplementary angles. We also know that <math>\angle 1 + \angle 4 = 180^\circ</math> because they too are supplementary. If that's the case, it must be true that <math>\angle 2 = \angle 4</math> by substitution. | ||
- | By the same logic, <math>\angle 1 = \angle 3</math> | + | By the same logic, <math>\angle 1 = \angle 3</math>. Angle 2 and angle 4 are vertical angles, and angle 1 and angle 3 are also vertical angles. |
[[File:Vertical_angles.gif|Vertical angles]] | [[File:Vertical_angles.gif|Vertical angles]] |
Revision as of 01:18, 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.
Vertical Angle Theorem
Vertical angles are equal
Look at the intersecting line segments EF & GH below. We know that since they're supplementary angles. We also know that because they too are supplementary. If that's the case, it must be true that by substitution. By the same logic, . Angle 2 and angle 4 are vertical angles, and angle 1 and angle 3 are also vertical angles.