What is a unique triangle in math

A unique triangle is a polygon that does not have any equivalent. It means that there is no other triangle that has the same dimensions and shape. Those trigonal figures are said to be unique that can be created in only one way. All triangles of this sort are congruent. Even if you turn or flip them to line up, the dimensions will remain the same, and just orientation will change. A trigon is perceived to be unique as long as the figure’s lengths are added up to be greater than the third.  

TRIANGLE

A triangle is a two-dimensional geometric figure. It has various types that can be classified based on different measurements and given information like sides and angles. There are some common characteristics of a triangle:

• These three-sided geometric figures are polygons.
• The three internal angles of the trigonal figure always add to 180 degrees. 

By sides, the triangle can be classified as equilateral, scalene, and isosceles. An equilateral triangle has three equal sides, isosceles have two sides of equal length, and scalene has no equal length side. Triangle can also be differentiated based on the angle, include acute, right-angle, and obtuse. If all internal angles of the trigon are less than 90, it is called acute. The trigonal figure that contains at least a 90-degree angle is known as the right-angle, and the obtuse trigon contains one angle greater than 90 degrees. Based on the given information, there are three ways to identify the figure:

• A Unique Triangle
• More Than One Triangle
• No Triangle

CONDITIONS TO CREATE AN UNIQUE TRIANGLE

There are certain conditions through which a unique trigonal figure can be constructed. If more than one trigonal figure can be constructed using the same measurements, this triangle will not be considered unique. 

Length of Three Sides

This condition is also known as the SSS triangle, stands for side-side-side. Having information on the length of all three sides of the figure tells you that it is unique. The sides’ length does not matter because even if the sides’ length is different, it will not affect the shape of the polygon. The missing angles can be figured out if one knows the lengths by using the law of cosine. For solving SSS, initially use the Law of cosine to calculate the first angle with A length. Then, again use the Law of cosine to find another angle by using B length. In the end, use the angles of a triangle add to 180° to find the last angle. This method will determine the angle but will not affect the figure.

Two Sides and Their Included Angle

This condition is abbreviated as SAS, which stands for side-angle-side. By using the SAS, one can construct a polygon. It is necessary to note that the side angle side will always create only one triangle called a unique trigon. You can flip up to down, rotate or move, but the shape will remain the same. The orientation of the figure does not change the size and angle, but only the direction. When one knows two sides and angles, we can find the other side and the remaining angle. We can find the unknown side by using the Law of cosine and Law of sine for the smaller of the other two angles. Finally, add the two angles to 180° to find the last angel.

Two Angles and Non-Included Side

This condition is called AAS, which stands for angle-angle-side. If two angles and the non-included length of a side are given, you can always create a distinctive polygon. This is known to be unique. The figure result of AAS will be congruent. The dimensions will not change by turning and flipping the figure. However, the directions will change, but not the length and angle. The AAS can be solved by finding the other angle by using the three angles add to 180°, and then the Law of sines can be used to find the other two sides. The solution of AAS is not so difficult and tricky.

Two Angles and Their Included Side

This fact is abbreviated as ASA, which stands for angle-side-angle. Knowing the two angles and the length of their included side will give a polygon that will not have any equivalent. ASA gives a distinctive triangle. By turning and flipping it to line up cannot affect its shape. The solution of ASA is almost similar to AAS. To find the third angle, you can use the three angles add to 180°. Then, by using the Law of sines, you can get the remaining two sides. 

IS SSA A UNIQUE TRIANGLE?

SSA stands for side-side-angle. Some people believe that SSA is a unique triangle because it also includes two sides and an angle just like SAS, but here is a thing to notice that the angle in SSA is non-included, whereas, in SAS, the angle is included. When we know two sides and the angle is non-included, it can construct the two triangles. SSA does not fulfill the requirement of a unique figure. That is why it cannot be included in the family of unique polygons.

DETERMINATION OF UNIQUE TRIANGLE

If one knows the conditions of distinct trigons, it will be easy to determine them. To the given triangle, we consider the conditions like measurements of sides and angles and the relationship between the sides and angle to determine if it is unique or not. By drawing a trigon under the three side conditions and two sides with an included angle, we will know that only one shape figure can be constructed under each condition, which fulfills the facts of a unique trigon.

CONCLUSION 

The triangle is a polygon figure which has three sides and can be classified on various terms. The trigon is unique and can be created only one way, and no other triangle of different shapes with the same measurements can be created. It is a little tricky to figure out, but if you know the unique triangle’s rules and conditions, you can determine it with a little effort.

Unique Triangle- Conditions to Create an Unique Triangle