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Adjacent Angles
From Latin: adjacere - "lie near"
Definition: Two angles that share a common side and a common vertex, but do not overlap
Try this Drag the orange dot. The line AC is the common leg of the two adjacent angles
In the figure above, the two angles BAC and CAD share a common side (the blue line segment AC). They also share a common vertex (the point A). They are therefore termed 'adjacent angles'.

Obviously, the larger angle BAD is the sum of the two adjacent angles.

They do not overlap

Two angles that overlap, one inside the other sharing a side and vertex In the figure on the right, the two angles PSQ and PSR overlap. Although they share a common side (PS) and a common vertex (S), they are not considered adjacent angles when they overlap like this. Adjacent angles must be next to each other, not one on top of the other.

Another way of defining them is: "two angles that share a side and a vertex, but do not share any interior points".

In Polygons

A polygon showing its interior angles, and a label pointing to two that are adjacent Another use of the term refers to the interior angles of polygons. Any two interior angles that share a common side are called the "adjacent interior angles" of the polygon, or just "adjacent angles". Here the word adjacent is used in its ordinary English meaning of "next to each other".
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Other angle topics

General

Angle Types

Angle relationships