You have sorted triangles by their sides and angles. Now a new question: given some measurements, can you build a triangle at all, and is there only one?
Key idea
Sides: three lengths make a triangle only if the two shorter ones add to more than the longest. Then there is exactly one triangle. Angles: three angles fit a triangle only if they add to $180^\circ$. Then there are many triangles: the same shape in different sizes.
Why it works
Lay the longest stick down. The other two are hinged at its ends and swing up. To make a corner, their free ends must meet above the long stick, and they can only reach each other if together they are longer than it. If they add to exactly the long side, they lie flat; if less, there is a gap. Angles alone cannot fix the size: a photo of a triangle, enlarged, has the same angles.
Worked examples
Worked example: It works
Can 4 cm, 5 cm and 8 cm be the sides of a triangle?
- The two shorter sides: $4 + 5 = 9$.
- $9 > 8$, so yes. The three lengths fix the triangle: exactly one triangle.
Answer: Yes, exactly one triangle
Worked example: It does not
Can 3 cm, 4 cm and 8 cm be the sides of a triangle?
- $3 + 4 = 7$, which is less than $8$.
- The short sides cannot reach each other: no triangle.
Answer: No triangle
Worked example: Range for the third side
Two sides of a triangle are 5 cm and 9 cm. What whole-number lengths can the third side have?
- It must be less than $5 + 9 = 14$.
- It must be more than $9 - 5 = 4$, or the 5 and the third side could not reach past the 9.
- So it is 5, 6, 7, …, 13 cm.
Answer: From 5 cm to 13 cm
Common mistake
Thinking any three lengths make a triangle, or that equal sums are enough ($3 + 5 = 8$ only lies flat). And three angles that add to $180^\circ$ do not give one triangle: they give a whole family of different sizes. Check sides by adding the two shortest; check angles by adding all three.