Triangle Inequality Rule
The Triangle Inequality Rule is a Grade 7 geometry concept in Big Ideas Math, Course 2 stating that the sum of any two sides of a triangle must be greater than the third side. For three lengths a, b, and c to form a triangle, all three conditions must hold: a + b > c, a + c > b, and b + c > a. For example, sides 5, 7, and 10 work since 5 + 7 = 12 > 10, but 3, 4, and 8 fail because 3 + 4 = 7 < 8. This rule determines whether a set of lengths can actually form a valid triangle and is used to find possible third-side lengths when two sides are given.
Key Concepts
For any triangle with side lengths $a$, $b$, and $c$, the sum of any two sides must be greater than the third side: $$a + b c$$ $$a + c b$$ $$b + c a$$.
Common Questions
What does the Triangle Inequality Rule state?
The sum of any two sides of a triangle must be strictly greater than the third side. This must hold for all three combinations of the three sides.
Do sides 5, 7, and 10 form a triangle?
Yes: 5+7=12>10, 5+10=15>7, 7+10=17>5. All three inequalities hold, so the triangle is valid.
Do sides 3, 4, and 8 form a triangle?
No: 3+4=7, which is less than 8. The first inequality fails, so these three lengths cannot form a triangle.
How do you find the range of possible values for a third side?
If two sides are a and b, the third side c must satisfy |a − b| < c < a + b. For sides 5 and 9: 4 < c < 14.
Why does the Triangle Inequality Rule work intuitively?
If two sides are too short to 'reach' across the third side, the ends cannot connect. The triangle collapses into a straight line when equality holds (degenerate case).
How many inequalities must you check to verify a set of three sides forms a triangle?
Three—one for each combination. However, if the sum of the two shorter sides exceeds the longest side, the other two inequalities are automatically satisfied.