A Detailed Proof about Congruent Angles

In this post, I want to explain in details how to prove about congruent angles. In the diagram below, $latex \angle 1$ and $latex \angle 2$ are complementary angles. Also, $latex \angle 3$ and $latex \angle 4$ are complementary angles. Now, if $latex \angle 1$ is congruent to $latex \angle 3$, prove that $latex \angle 2$ is congruent to angle $latex \angle 4$.

congruent angles


From the problem above, we have the following facts (given).

$latex \angle 1$ and $latex \angle 2$ are complementary
$latex \angle 3$ and $latex \angle 4$ are complementary
$latex \angle 1 \cong \angle 3$

We want to prove that $latex \angle 2 \cong \angle 4$. Continue reading

5 Help Tools for Learning Mathematical Proofs

Before proving a theorem, mathematicians must be able to find something to prove. They must see patterns, explore them, test for many cases, and make conjectures and generalization. The final part of the process is the proof.

Mathematical Proofs - GeoGebra

Mathematicians during the early times rely only pencil and paper in investigating patterns. Today, hundreds of free and commercial software are available and accessible, even for high school students like you. Below are some of the free programs that you can use to explore patterns and improve your skills in mathematical proofs. Use these programs to investigate future problems here in Proofs from the BookContinue reading

What are mathematical proofs, really?


In an elementary class, the teacher asked the pupils to do an experiment. He told them to plant flowers and group the flowers into two.  The first group was to be placed outdoors where there is sunlight. The other group was to be kept in a dim room. After a while, the pupils observed that the flowers outdoors grew healthy, while those kept indoors either died or were not as healthy. The pupils concluded that plants need sunlight.

flowers and mathematical proofs

In the experiment, the pupils observed that flowers with enough sunlight are healthier compared to those placed indoor. Some pupils may have an idea (a hypothesis) about this concept based on prior observation. The conclusion that plants need sunlight was based on the observation from the experiment.

In mathematics, similar situations occur. The hypotheses in mathematics are based on observed patterns. For instance, we can “conclude” from a few examples that the sum of two even integers is even. Continue reading