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How to check whether three measurements make a triangle

Three tests: side inequality, angle sum, and enough information

For three side lengths, sort them as $a \leq b \leq c$ . They form a nondegenerate triangle exactly when $a + b > c$ . For three angles, all must be positive and their sum must be $18 0^{\circ}$ . If all three side lengths are fixed, SSS determines one triangle up to reflection, so all such triangles are congruent. Three angles alone determine shape but not size, so AAA does not guarantee congruence.

Problem

Decide whether each set can form a triangle. If it can, decide whether every triangle with those measurements must be congruent. (a) Side lengths $4 cm, 6 cm, 9 cm$ . (b) Angle measures $5 1^{\circ}, 1 5^{\circ}, 2 4^{\circ}$ . (c) Side lengths $12 m, 38 m, 14 m$ .

Answer

(a) Yes, and all such triangles are congruent. (b) No. (c) No.

Checked independently with the triangle inequality, angle-sum theorem and SSS.

Step-by-step solution

(a) Sides 4, 6 and 9

Only the two shorter sides need the decisive check:

4 + 6 = 10 > 9

So the lengths form a triangle. Because all three sides are specified, SSS fixes the triangle, and every triangle with these side lengths is congruent.

(b) Angles 51°, 15° and 24°

Angles of a triangle must total $18 0^{\circ}$ :

5 1^{\circ} + 1 5^{\circ} + 2 4^{\circ} = 9 0^{\circ}

The sum is not $18 0^{\circ}$ , so these cannot be the three angles of a triangle.

(c) Sides 12, 38 and 14

The longest proposed side is $38$ . Add the other two:

12 + 14 = 26 < 38

They cannot meet to close a triangle, so no triangle exists.

Common traps

Equality is not enough: $a + b = c$ makes a flat segment, not a triangle. Also, knowing three valid angles does not force congruence - triangles can have the same angles at different scales.

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