Triangles form one of the most significant and large parts of geometry for a good purpose. There are a lot of concepts in triangles themselves, from the most basic to the most advanced concepts. The trigonometry branch of mathematics is itself based on triangles and that is enough to prove how important triangles are. We all would have heard of equilateral triangles, isosceles triangles, and scalene triangles as they are the most basic classification of triangles. But did you know there was something called unique triangles? Here we will see about Unique Triangle.
Unique Triangle
The word unique means one of a kind. This is the same for unique triangles. Unique triangles are those triangles for which there exists no other triangle with the same dimensions or shape. As the name signifies, this is a triangle that is unique, that is, all its duplicates are congruent with each other. These types of triangles can be formed in a variety of methods. This is quite an easy concept in mathematics and can be fun to learn too!
We will learn more about unique triangles in this article!
Conditions for a unique triangle
Typically, with given measurements, there can be majorly three outcomes in drawing a triangle. Those three outcomes are either getting a unique triangle, getting no triangle at all, or getting multiple triangles with the same measurements. There are some conditions to be able to draw a unique triangle.
What are the measurements that are needed to be able to draw a unique triangle? This is the one question that strikes all brains when talking about the concept of unique triangles. Read further on to learn all the conditions for a unique triangle.
SSS (Side-Side-Side)
As the name mentions, when the dimensions of all three sides of a triangle are given, a triangle is drawn and this triangle is like none else. There can be duplicates or flipped and rotated versions, but eventually, they are all duplicates of one another. They are all congruent to one another so qualify as a single triangle, making it a unique triangle. These triangles, drawn with the same side measurements satisfy the SSS condition of congruence.
But this is possible provided that the inequality of triangles is met. That is, the sum of any two sides of the triangle should always be greater than the third side. Else, the given measurements of line segments will not be able to form a triangle.
SAS (Side-Angle-Side)
When the lengths of two sides and the measure of the angle between the two line segments are given, a unique triangle can be formed. The angle given should be the angle formed by the two sides and not any other angle in the triangle. The mentioned angle, between the two sides, is also called the “included” angle. The triangle formed thus will be unique and just like in the case of a triangle formed with SSS, all the other triangles formed like this are mere duplicates of each other and will be congruent to one another. There can be a lot of triangles formed with the same measurements but you can be sure that if such triangles are moved, flipped, or rotated, they will be identical to one another. Triangles of these types can satisfy the SAS condition of congruence.
ASA (Angle-Side-Angle)
When the measure of two angles and the length of the included side is given to you, you can form a triangle that is unique from all the other triangles. The measure of two angles should be given and the length of the side given has to be an included side, that is, the given side should be in between the two given angles only. You can draw many variations of the same triangle but they are all identical to each other. They also satisfy the ASA condition of congruency and are thus, congruent.
AAS or SAA (Angle-Angle-Side or Side-Angle-Angle)
Just like the other conditions, this signifies two angles and one side or one side and two angles in the same order. The measures of two angles and the measure of a side that is given have to be in the same order, that is, the side given should be an immediate follower or predecessor of the two given angles. The side should not be an included one, but rather some other side. This means that the side should not be in between the given angles, but rather one of the other two sides. If the side is in between the two given angles, the condition would become one of ASA rather than of AAS or SAA.
HL (Hypotenuse-Leg)
This is applicable only for right-angled triangles because a hypotenuse is involved. The hypotenuse is the side opposite to the right angle in a triangle and the leg is one of the sides that forms the right angle. A unique triangle can be formed with the hypotenuse and leg as it eventually becomes a condition that falls under SSA. Just like all the other unique triangles, multiple triangles can be formed with the given measurements but all of them will be congruent to one another and thus are merely duplicates of each other.
Conditions that do not form a unique triangle
AAA (Angle-Angle-Angle)
If three angles are given this is not enough to form a unique triangle as the same angle measurements can be used to form triangles of different sizes. This is not enough to form a unique triangle. Triangles with sides of various measurements can be formed with the same angle measures but all these triangles will be similar to each other, just not congruent.
SSA or ASS (Side-Side-Angle or Angle-Side-Side)
They essentially mean the same but are not enough to form a unique triangle. One can form two triangles that are not congruent to each other. They can form two triangles and every triangle thus formed is congruent to either of these two triangles.
Conclusion
A unique triangle is a triangle that has no other triangle that can be formed with the same set of conditions. There are a set of ways to form unique triangles and this is a very simple concept in triangles, which has a lot of concepts in it. You can try to take different values and try to draw different triangles.
FAQs
- Are all triangles unique?
No, not all triangles are unique but most of the triangles that you draw will be congruent with one another.
- What is a unique triangle with an example?
A unique triangle is a triangle that does not have any other triangle resembling it. For example, a triangle formed with sides 8 cm, 7 cm, and 9 cm is unique.
