An angle is formed by two rays that share a common endpoint called the vertex. Angles are measured in degrees, with a full circle being 360 degrees. There are several types of angles based on their measure: acute angles are less than 90 degrees, right angles are exactly 90 degrees, obtuse angles are between 90 and 180 degrees, and straight angles are exactly 180 degrees, forming a straight line.
Two angles are supplementary if the sum of their measures is exactly 180 degrees. For example, a 60 degree angle and a 120 degree angle are supplementary because 60 plus 120 equals 180. Similarly, a 45 degree angle and a 135 degree angle are supplementary. It's important to note that supplementary angles do not need to be adjacent or share a common vertex. They can be separate angles anywhere, as long as their measures add up to 180 degrees.
When two angles are both adjacent and supplementary, they form what's called a linear pair. Adjacent angles share a common vertex and a common side, but have no interior points in common. In a linear pair, the non-common sides of the two angles form a straight line. This means that the sum of the measures of these two angles must be 180 degrees, since they together form a straight angle. For example, if we have a straight line with point O in the middle, and two rays extending from O, the angles on either side of point O are adjacent supplementary angles, forming a linear pair.
We can use the supplementary angle relationship to find unknown angle measures. Let's work through two examples. First, if we need to find the supplement of a 65 degree angle, we set up the equation: 65 degrees plus x equals 180 degrees. Solving for x, we subtract 65 from both sides to get x equals 115 degrees. For our second example, suppose two supplementary angles are in the ratio 2 to 3. We let the angles be 2x and 3x. Since they're supplementary, their sum is 180 degrees, so we have 2x plus 3x equals 180 degrees. Combining like terms gives us 5x equals 180 degrees. Dividing both sides by 5, we find x equals 36 degrees. Therefore, the two angles are 2 times 36 equals 72 degrees and 3 times 36 equals 108 degrees.
Supplementary angles have many practical applications in the real world. In architecture, roof trusses often form supplementary angles to distribute weight evenly. The angles in a typical roof truss add up to 180 degrees, creating a stable structure. In engineering, bridge supports use supplementary angles to ensure structural integrity. The diagonal supports in a bridge form supplementary angles with the vertical supports, providing maximum strength. Even everyday objects like scissors demonstrate supplementary angles. When scissors are partially open, the angles between the blades and the handles are supplementary, adding up to 180 degrees. Understanding these relationships helps architects, engineers, and designers create stable and functional structures.
Let's practice with some problems on supplementary angles. Problem one: If two angles are supplementary and one is 35 degrees, what is the measure of the other? Since supplementary angles add up to 180 degrees, we subtract 35 from 180 to get 145 degrees. Problem two: Two supplementary angles are in the ratio 4 to 5. Find the measures of both angles. Let the angles be 4x and 5x. Since they're supplementary, 4x plus 5x equals 180 degrees. This gives us 9x equals 180 degrees, so x equals 20 degrees. Therefore, the angles are 4 times 20 equals 80 degrees and 5 times 20 equals 100 degrees. Problem three: In a diagram, angles AOB and BOC are supplementary. If angle AOB equals 2x plus 10 degrees and angle BOC equals 3x minus 5 degrees, find x. Since they're supplementary, their sum is 180 degrees. So we have the equation: 2x plus 10 plus 3x minus 5 equals 180. Simplifying, we get 5x plus 5 equals 180. Subtracting 5 from both sides gives 5x equals 175. Dividing by 5, we find x equals 35 degrees.