How To Prove Two Triangles Are Congruent 

Mathematics being a key subject in academics, nothing should be neglected if success is to be attain. Every topic has its own importance, as well as applications. Which means each topic ought to be taken seriously. 

Although there are topics that are quite technical and difficult to understand. But with consistent practise, it becomes easy. I know all these because I have passed through the process and I know what it feels like to study mathematics. 

One of the topics that gave me hard time was learning how to prove two triangles are congruent. But when I did, solving questions related to congruent triangles became rather too easy. To aid the learning process for other mathematics students out there, I have decided to share how I was able to learn how to prove two triangles are congruent. 

First of all,  let’s try to understand what congruence is all about before learning how to prove two triangles are congruent. 

What is congruence? 

This is a mathematical term used to refer to when two separate triangles have their corresponding sides and angle equal to each other. When such occurs, these two triangles are said to be congruent of each other. 

Having understood the basics of congruence and what it is all about, I think it’s time to know how it can be proved. Understand this can take quite tricky, but with the I will be breaking down the proving into simple terms, anyone going through this should be able to understand it perfectly well. 

How is congruent of two triangles proved? 

While trying to understand the basics of congruent of two triangles in order to help improve my knowledge of how it solved, I came about five different theorems which are, Side-Side-Side (SSS) Theorem, Side-Angle-Side (SAS) Theorem, Angle-Side-Angle (ASA) Theorem, Angle-Angle-Side (AAS) Theorem, and Hypotenuse-Leg (HL) Theorem. With these theorems, I was able to understand how it is proved. 

To also help others learn, I will be explaining what I was able to understand from these theorems each at a time; 

  • Side-Side-Side (SSS) Theorem – This was the first way to prove two triangles are congruent that I discovered. Here, the three sides of the two triangles have to be proven to be congruent. 
  • Side-Angle-Side (SAS) Theorem – This particular theorem is another way to prove this. To prove this, two corresponding sides of the triangle and an angle which is inbetween the sides are congruent. 
  • Angle-Side-Angle (ASA) Theorem – Another way to understand this is that two angles and one side which is in between the angles of the two triangles has to be congruent. With this, these two triangles can be considered congruent. 
  • Angle-Angle-Side (AAS) Theorem – For this theorem to hold, two corresponding angles of two triangles and one side which is not in between the two angles have to be congruent. 
  • Hypotenuse-Leg (HL) Theorem – This is the last theorem and it is quite different from others. For this hold, the two triangles must be a right angle triangle and they must be the same in sides. Which means they both should have the same hypotenuse and sides..

Lastly, I also took note of two other ways of proving two triangles congruent that totally wrong. They are r Side-Side-Angle (SSA) and Angle-Side-Side (ASS). These two are not applicable, which implies that they shouldn’t be used.