Many high textbooks consider the congruence theorems (SSS Congruence Theorem, SAS Congruence Theorem, ASA Congruence Theorem) as postulates. This is because their proofs are complicated for high school students. However, let us note that strictly speaking, in Euclidean Geomtery (the Geometry that we learn in high school), there are only five postulates and no others. All of other postulates mentioned in textbooks aside from these five are really theorems without proofs.
In this post, we are going to prove the SSS Congruence Theorem. Recall that the theorem states that if three corresponding sides of a triangle are congruent, then the two triangles are congruent.
Before proving the SSS Congruence theorem, we need to understand several concepts that are pre-requisite to its proof. These concepts are isometries particulary reflection and translation, properties of kites, and the transitive property of congruence. If you are familiar with these concepts, you can skip them and go directly to the proof. Continue reading
The Pythagorean Theorem is probably the most popular theorem in school mathematics. Surely, you have heard or read about it at least once from elementary school to high school. The Pythagorean Theorem states that given a right triangle with shorter sides , , and hypotenuse , the following equation holds
Math tricks and mental math shortcuts are used to perform speedy calculations. Many of us know at least a trick or two because they are easy to remember and handy to use. But why or how do these tricks work?
This is the first of a series of posts that will discuss math tricks and mental math shortcuts. Aside from the algorithm that will be taught in this series that will benefit everyone, we will also investigate why they work.
The Trick: Squaring Numbers Ending in 5
Squaring numbers ending in 5 is probably the most popular among the math tricks and mental shortcuts. Below is the algorithm of the trick.
- Take 25 as the last two digits of the product of the factors.
- Add 1 to the tens digit of one of the factors.
- Take away 5 from both the factors, and multiply the remaining numbers.
- Append the product in the third step to the left of the result in the first step. Continue reading