Lemma: In any triangle, the sum of the interior angles is always 180 degrees.
Why it's a lemma:
* It's a smaller, less important result. While the fact that the angles of a triangle sum to 180 degrees is fundamental in geometry, it's often used as a stepping stone to prove larger theorems.
* It's used to prove a bigger theorem. This lemma is crucial for proving the Pythagorean Theorem, for instance, as it's used to show that the angles in a right triangle relate in a specific way.
Example of using the lemma:
Theorem: The angles of a quadrilateral sum to 360 degrees.
Proof:
1. Divide the quadrilateral into two triangles. This can be done by drawing a diagonal across the quadrilateral.
2. Apply the lemma: Each triangle has angles that sum to 180 degrees.
3. Combine the results: The total angle sum of the quadrilateral is the sum of the angles of the two triangles, which is 180 + 180 = 360 degrees.
In this example, the lemma about triangle angle sum is used as a key step in proving the larger theorem about quadrilateral angle sum.
Note: Lemmas can be used in various areas of mathematics, not just geometry. They are often simple statements that are proven and then used as building blocks for more complex proofs.
