Constructing Triangles & The Triangle Inequality Theorem - Seventh Grade Mathematics

The geometric construction of triangles is governed by strict mathematical conditions, where side lengths must satisfy the Triangle Inequality Theorem and angle measures must sum precisely to 180 degrees.

The Triangle Inequality Theorem

Not every combination of three line segments can close to form a valid triangle. For three segments to meet and enclose a triangular region, the combined length of any two segments must be strictly greater than the length of the remaining segment.

               THE TRIANGLE INEQUALITY THEOREM
   For any triangle with side lengths a, b, and c:
       a + b > c       a + c > b       b + c > a

If the sum of any two sides is less than or equal to the third side, the two shorter segments cannot bridge the span across the longest side, and no triangle can be constructed.

CASE 1: a + b < c (Cannot Connect)
   Segment c:  [========================================] (10 cm)
   Segment a:  [========] (3 cm)
   Segment b:             [============] (4 cm)
   Gap of 3 cm remains; segments cannot reach each other.

CASE 2: a + b = c (Degenerate Segment - Flat Line)
   Segment c:  [========================================] (10 cm)
   Segment a:  [================] (4 cm)
   Segment b:                   [=======================] (6 cm)
   Segments collapse flat onto segment c; no triangular interior formed.

CASE 3: a + b > c (Valid Triangle)
   The segments swing upward and meet at a unique third vertex.

The Quick Test for Valid Side Lengths

To determine whether three given side lengths can form a triangle, you do not need to test all three inequalities. It is sufficient to test only the two shortest sides: if the sum of the two smaller sides is strictly greater than the longest side, the three segments form a triangle.

Example A: Side lengths 4 cm, 7 cm, 12 cm.
Shortest sides: 4 + 7 = 11. Longest side: 12.
Since 11 is NOT greater than 12, NO triangle can be formed.

Example B: Side lengths 6 cm, 8 cm, 11 cm.
Shortest sides: 6 + 8 = 14. Longest side: 11.
Since 14 > 11, a valid triangle CAN be formed.

Determining the Range of the Third Side

When you are given two fixed side lengths a and b, the Triangle Inequality Theorem defines the exact open interval of possible lengths for the unknown third side c.

               POSSIBLE RANGE FOR THIRD SIDE c
                     |a - b| < c < a + b

The third side must be strictly greater than the absolute difference of the two known sides, and strictly less than their sum.

Given sides: 7 cm and 10 cm.
Difference: 10 - 7 = 3 cm
Sum:        10 + 7 = 17 cm
Valid Range for Third Side c:  3 cm < c < 17 cm

Any length between three centimeters and seventeen centimeters (exclusive) creates a valid triangle.

Triangle Construction Conditions: Unique, Multiple, or None

Geometric conditions determine whether zero, exactly one (unique), or infinitely many triangles can be constructed.

+-----------------------+-----------------------------+------------------------+
| Given Conditions      | Geometric Designation       | Number of Triangles    |
+-----------------------+-----------------------------+------------------------+
| 3 Valid Side Lengths  | SSS (Side-Side-Side)        | Exactly ONE Unique     |
| 2 Sides + Included Ang| SAS (Side-Angle-Side)       | Exactly ONE Unique     |
| 2 Angles + Included Sd| ASA (Angle-Side-Angle)      | Exactly ONE Unique     |
| 3 Angle Measures      | AAA (Angle-Angle-Angle)     | INFINITELY Many        |
| a + b <= c            | Violation of Inequality     | NONE (Zero)            |
| Angle sum != 180 deg  | Violation of Angle Sum      | NONE (Zero)            |
+-----------------------+-----------------------------+------------------------+

When three angle measures summing to one hundred eighty degrees are specified (such as 40 degrees, 60 degrees, and 80 degrees), infinitely many similar triangles of different overall sizes can be constructed. However, fixing three side lengths (SSS) locks the triangle into an immutable, rigid structure, yielding exactly one unique triangle.

Exercises

Work through each problem carefully in your notebook before reading the provided solutions.

Exercise 1. Determine whether a triangle can be constructed with side lengths measuring five inches, eight inches, and thirteen inches. Justify your conclusion using the Triangle Inequality Theorem.

Exercise 2. Two sides of a triangle measure nine meters and fifteen meters. Write an inequality expressing the full range of possible lengths for the third side x.

Exercise 3. A student is given three angle measures: fifty degrees, sixty-five degrees, and sixty-five degrees. Can a triangle be constructed with these angles? If so, does it produce a unique triangle or infinitely many triangles?

Exercise 4. Is it possible to construct a triangle with angles measuring forty-five degrees, seventy degrees, and seventy-five degrees? Explain your reasoning.

Solutions

Solution to Exercise 1. Test the two shorter sides: five inches plus eight inches equals thirteen inches. For a triangle to exist, the sum of the two shorter sides must be strictly greater than the longest side. Here, five plus eight equals thirteen, which is equal to, not greater than, the longest side of thirteen inches. The segments would collapse into a flat line segment of thirteen inches with zero height. Therefore, no triangle can be formed.

Solution to Exercise 2. The length of the third side x must be strictly greater than the difference between the two known sides and strictly less than their sum. Find the difference: fifteen minus nine equals six meters. Find the sum: fifteen plus nine equals twenty-four meters. The range of possible lengths for the third side is six meters is less than x is less than twenty-four meters (6 < x < 24).

Solution to Exercise 3. First check the sum of the three angle measures: fifty degrees plus sixty-five degrees plus sixty-five degrees equals one hundred eighty degrees. Because the sum is exactly one hundred eighty degrees, a triangle can indeed be constructed. However, when only three angle measures are given without any fixed side lengths (AAA condition), infinitely many similar triangles can be constructed, ranging from tiny figures to massive figures sharing the exact same shape.

Solution to Exercise 4. Add the three given angle measures: forty-five plus seventy plus seventy-five. Forty-five plus seventy equals one hundred fifteen; one hundred fifteen plus seventy-five equals one hundred ninety degrees. The interior angles of any triangle in Euclidean plane geometry must sum to precisely one hundred eighty degrees. Because the sum of these angles is one hundred ninety degrees, it is impossible to construct a triangle with these angles.