Triangles have rules about their sides and angles that help us understand their shape. The Triangle Inequality Theorem tells us how the sides relate, while angle-side relationships show how angles and sides correspond.

Indirect proofs are a clever way to prove things about triangles by assuming the opposite. We can use these methods to figure out if certain triangles are possible and how their parts connect.

Triangle Inequalities

Triangle Inequality Theorem application

• States sum of any two side lengths must exceed third side length
  • For triangle sides $a$, $b$, and $c$:
    • $a + b > c$
    • $b + c > a$
    • $c + a > b$
• Difference between any two side lengths must be less than third side length
  • For triangle sides $a$, $b$, and $c$:
    • $|a - b| < c$
    • $|b - c| < a$
    • $|c - a| < b$
• Check all combinations of two side lengths sum greater than third length to determine if triangle possible (5, 7, 10)

Third side range in triangles

• Triangle Inequality Theorem determines range for third side given two known side lengths
  • Third side greater than absolute difference of known sides
  • Third side less than sum of known sides
• For triangle with sides 5 and 8, third side $x$ range:
  • $|5 - 8| < x < 5 + 8$
  • $3 < x < 13$

Indirect Proofs and Triangle Relationships

Indirect proofs for triangles

• Proves statement by assuming opposite and showing contradiction
• Indirect proof steps:
  1. Assume opposite of statement to prove
  2. Use logic and given info to reach contradiction
  3. Original statement true since assumption contradicts
• Proves triangle statements like certain types impossible or side/angle relationships must hold

Angle-side relationships in triangles

• Longest side opposite largest angle, shortest side opposite smallest angle
• Congruent sides have congruent opposite angles (isosceles)
• Congruent angles have congruent opposite sides (isosceles)
• Sum of three angle measures always 180°
• Pythagorean Theorem for right triangles: sum of leg lengths squared equals hypotenuse length squared
  • For legs $a$ and $b$ and hypotenuse $c$:
    • $a^2 + b^2 = c^2$