what is the relationship between a linear pair and supplementary angles
The correct answer and explanation is:
A linear pair and supplementary angles are closely related concepts in geometry, and understanding one can help in understanding the other.
A linear pair of angles consists of two adjacent angles that are formed when two lines intersect. These angles share a common vertex and a common arm. The key property of a linear pair is that the sum of the two angles always equals 180 degrees. This happens because the two angles together form a straight line, which has a measure of 180 degrees.
Supplementary angles, on the other hand, are two angles whose measures add up to 180 degrees, but they do not have to be adjacent or formed by two intersecting lines. In other words, supplementary angles can be located anywhere as long as their sum equals 180 degrees. The concept of supplementary angles applies more generally, not just to adjacent angles, but also to angles that might be far apart from each other.
The relationship between a linear pair and supplementary angles is that every linear pair of angles is a pair of supplementary angles. This is because the two angles in a linear pair always add up to 180 degrees. However, not all supplementary angles are linear pairs. Supplementary angles can exist in situations where the angles are not adjacent to each other, like when two non-adjacent angles are on a straight line but not formed by intersecting lines.
In summary, while both linear pairs and supplementary angles add up to 180 degrees, the main difference lies in the specific arrangement of the angles. A linear pair requires the angles to be adjacent and formed by two intersecting lines, whereas supplementary angles can exist in various configurations as long as their sum equals 180 degrees.