Final answer:
A linear pair is a pair of adjacent angles whose non-common sides form a straight line, and they must add up to 180 degrees, thus are supplementary. However, supplementary angles do not need to be adjacent; they just need to sum up to 180 degrees, hence not all supplementary angles form a linear pair.
Step-by-step explanation:
All linear pairs are supplementary angles because by definition, a linear pair consists of two adjacent angles formed when two lines intersect, creating angles that share a common side and whose non-common sides form a straight line. This straight line implies that the two angles add up to 180 degrees, which is the definition of supplementary angles. However, not all supplementary angles are linear pairs. Supplementary angles simply need to sum up to 180 degrees, but they do not have to be adjacent or form a linear pair. They could be separated in space, or be part of different geometric configurations, such as angles inside a polygon.
For example, in a triangle, which is a three-sided figure lying on a plane, the three interior angles add up to 180 degrees, and any two of these angles can be supplementary if they add up to 180 degrees when taken together. However, these angles do not form a linear pair because they do not sit on the same straight line.
The statement also raises questions about the angle relationships in geometric figures like triangles and the nature of vectors. Specifically, vector addition and the techniques for finding resultant vectors when vectors are at an angle to each other, which typically requires geometric and trigonometric methods.
This statement is true because supplementary angles are defined by their sum, not their arrangement. Two angles that add to 180 degrees are supplementary whether they're adjacent and forming a linear pair or not. Conversely, a linear pair is automatically supplementary because the two adjacent angles form a straight line, which, by geometric necessity, must sum to 180 degrees.