Theorems
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==SAS Triangle Congruence Theorems== | ==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. | + | ===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''. | ''Important note: The included angle is the angle between the two corresponding sides''. |
Revision as of 03:29, 22 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 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 , 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).
Supplementary Angle Theorem
Two angles that add up to 180 degrees are said to be supplementary angles
Complementary Angle Theorem
Two angles that add up to 90 degrees are said to be complementary angles
The Sum of the Interior Angles of a Triangle
The sum of the interior angles of a triangle are 180 degrees
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.
SSS Triangle Congruence Theorems
If 3 sides of one triangle are equal in length to 3 sides of another triangle, the two triangles are congruent. This is called the Side-Side-Side triangle congruence theorem, or the SSS theorem.
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: