Adjacent angles

angleA angle it is the part of the plane comprised between two rays that have a common origin. The rays are called sides and the common origin is vertex.

Two angles are adjacent if they have a side, the vertex in common and are supplementary.

Definition

The adjacent angles they are those that have one side and a vertex in common, in addition to that their other two sides are opposite rays. Two adjacent angles are both consecutive and supplementary, because together they are equal to a straight angle (180⁰).

adjacent angles

The angle α is adjacent to β because: 1) they have one side in common (the line CB), 2) the vertex in common (point B) and 3) both are consecutive and add up to 180⁰, that is, they are supplementary.

NOT adjacent angles

A and B they are not adjacent. They share the vertex and one side, but they are not supplementary. This is a typical error, sometimes the slip is made to name adjacent consecutive angles.

Usually, two angles are adjacent if:

  • They have a common side.
  • They have the vertex in common.
  • They are consecutive.
  • They are supplementary (their sum results in 180⁰).

Properties

  • The breasts of the adjacent angles are the same, example:
    • sin α = sin (180⁰ - α)
    • sin α = sin (π - α)
    • sin 120⁰ = sin 60⁰
  • The cosines of the adjacent angles are of equal absolute value, but of inverse sign, example:
    • cos α = - cos (180⁰ - α)
    • cos α = - cos (π - α)
    • cos 120⁰ = - cos 60⁰