Section 5-5 Inequalities in triangles GEOMETRY

Comparison Property of Inequality If 𝑎=𝑏+𝑐 and 𝑐>0, then 𝑎>𝑏. Given: 𝒂=𝒃+𝒄,𝒄>𝟎 Prove: 𝒂>𝒃 Statements Reasons 1.) 2.) 3.) 4.) 5.)

Theorem 5-10: If two sides of a triangle are not congruent, then the larger angle lies opposite the longer side. 𝑰𝒇 𝑿𝒁>𝑿𝒀, 𝒕𝒉𝒆𝒏 𝒎∠𝒀>𝒎∠𝒁.

Example 1: You are building a deck and want to put benches in the two largest corners, but which corners are the largest?

Theorem 5-11: If two angles of a triangle are not congruent, then the longer side lies opposite the larger angle. 𝑰𝒇 𝒎∠𝑨>𝒎∠𝑩, 𝒕𝒉𝒆𝒏 𝑩𝑪>𝑨𝑪.

Example 2: Which side is shortest?

Theorem 5-12 (Triangle Inequality Thm): The sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Example 3:

Example 4: The lengths of two sides of a triangle are given. Describe the lengths possible for the third side. A. B.