Side Lengths of Triangles

General Definition: For a triangle to be formed, the sum of any two side lengths must be greater than the length of the third side.
Unique Triangle Condition: If the line segments satisfy the conditions of the Triangle Inequality Theorem, then exactly one unique triangle is formed.
Mathematical Range for Unknown Sides: Given a triangle $ABC$ with known side lengths of $AB = 7\,cm$ and $BC = 9\,cm$ , the third side $AC$ can be determined by the following constraints:
- Calculation logic: $AC < 9 + 7$ and $9 < AC + 7$
- Combined inequality: $2 < AC < 16$
- The greatest value $AC$ could be is less than $16\,cm$ .
- The shortest value $AC$ could be is greater than $2\,cm$ .

Side-Based Classification: Triangles are categorized by their side lengths into three types:
- Equilateral: All three sides are equal in length.
- Isosceles: At least two sides are equal in length.
- Scalene: All three sides have different lengths.
Side-Angle Relationship: There is a direct relationship between the measure of internal angles and the length of the opposite sides:
- Longest Side: In any triangle, the longest side is always opposite the largest angle.
- Shortest Side: In any triangle, the shortest side is always opposite the smallest angle.
Opposite Side Identification Practice (Triangle ABC):
- Side $BC$ is opposite angle $A$ .
- Side $AC$ is opposite angle $B$ .
- Side $AB$ is opposite angle $C$ .

Triangle MNO:
- Given angles: $\angle O = 114.5^\circ$ and $\angle M = 37^\circ$ .
- Calculated missing angle: $\angle N = 180^\circ - (114.5^\circ + 37^\circ) = 28.5^\circ$ .
- Greatest side identification: Side $MN$ is the greatest because it is opposite $\angle O$ , which is the largest angle.
Triangle STU:
- Given angles: $\angle S = 62^\circ$ and $\angle U = 56^\circ$ .
- Calculated missing angle: $\angle T = 180^\circ - (62^\circ + 56^\circ) = 62^\circ$ .
- Shortest side identification: Side $SU$ is the shortest because it is opposite $\angle T$ , the smallest angle (per internal keys identifying $\angle T$ as the reference for the shortest side).
Triangle VWX:
- Internal angle values provided for the set: $61.3^\circ$ , $90^\circ$ , and $28.7^\circ$ (inferred from diagrams and keys).
- Ordering of side lengths from least to greatest: $WX$ , $VW$ , $VX$ .

Case Study 1: Side lengths of $13$ , $7$ , and $5$ units.
- Verification 1: $5 + 7 < 13$
- Verification 2: $13 + 5 > 7$
- Verification 3: $7 + 13 > 5$
- Conclusion: These segments do not form a triangle because the sum of the two shorter sides ( $5+7=12$ ) is not greater than the third side ( $13$ ).
Case Study 2: Side lengths of $4$ , $6$ , and $8$ units.
- Verification 1: $4 + 6 > 8$
- Verification 2: $6 + 8 > 4$
- Verification 3: $8 + 4 > 6$
- Conclusion: These segments form a triangle because the sum of any two side lengths is greater than the length of the third side.

Using measured line segments (nearest inch), the following combinations are evaluated for triangle formation:

Segments A, B, D: Result: Yes. The sum of any two side lengths is greater than the length of the third.
Segments A, E, C: Result: No. The sum of lines $A$ and $C$ is not greater than line $E$ .
Segments A, C, D: Result: Yes. The sum of any two side lengths is greater than the length of the third.
Segments B, F, E: Result: No. The sum of lines $B$ and $F$ is not greater than line $E$ .
Segments D, B, F: Result: No. The sum of lines $D$ and $F$ is not greater than line $B$ .

The Straw Theorem Application: In a scenario where three straws measure $10\,inches$ , $12\,inches$ , and $18\,inches$ :
- Student Logic (Ryan): Contends there is only one unique way to arrange them.
- Student Logic (Marcy): Contends there are many ways to arrange them.
- Conclusion: Ryan is correct. According to the properties of geometry, given three specific side lengths that satisfy the Triangle Inequality Theorem, those lengths will form exactly one unique triangle. The straws cannot be arranged in multiple ways to produce different triangular shapes.