Geometry / Unit 6: Relationships in Triangles
6.5 Inequalities in One Triangle
Notes
Sides and angles are linked: bigger angles face bigger sides.
- The longest side is opposite the largest angle; the shortest side is opposite the smallest angle.
- Triangle Inequality Theorem: the sum of any two sides must be greater than the third side.
- Third side range: if two sides are $a$ and $b$, then $|a - b| < c < a + b$.
- Example: sides 4 and 9. The third side is between 5 and 13 (not including either).
Practice
Which three lengths can form a triangle?
$6 + 7 = 13 > 12$. The others fail: $3 + 4 < 8$, $5 + 5 = 10$, $2 + 9 < 12$.
Two sides of a triangle are 8 and 15. Which could be the third side?
The third side must be between $15 - 8 = 7$ and $15 + 8 = 23$, not including them.
In $\triangle ABC$, $m\angle A = 80^\circ$, $m\angle B = 45^\circ$. Which side is longest?
$m\angle C = 55^\circ$, so $\angle A$ is largest, and the side opposite it, $\overline{BC}$, is longest.
Ready for more? Take the Unit 6 test.