Question

An illustration shows three rays from a common vertex with labeled angles. The first ray is inclined to the horizontal left of the common vertex with an arrow at the end. The second ray is inclined to the horizontal right of the common vertex with an arrow at the end. The third ray is to the top right of the common vertex with an arrow at the end. It is between the other two rays. The angle between the first and the third rays is labeled as left parenthesis 2 x plus 3 right parenthesis degrees. The angle between the second and third rays is labeled as left parenthesis x minus 6 right parenthesis degrees.
What is an equation for these two adjacent angles?

Answer 1

To find an equation for the two adjacent angles described in the illustration, we can use the property that the sum of angles around a point is equal to 360 degrees. Given that the angle between the first ray (leftward) and the third ray (top right) is labeled as \((2x + 3)^\circ\) and the angle between the second ray (rightward) and the third ray (top right) is labeled as \((x - 6)^\circ\), we can set up the following equation:

1. The sum of the two angles we are interested in is 180 degrees because they are adjacent angles forming a straight line with the angle between them (the third ray):

\[
(2x + 3) + (x - 6) = 180
\]

2. Now we simplify and solve the equation:

\[
2x + 3 + x - 6 = 180
\]
\[
3x - 3 = 180
\]
\[
3x = 180 + 3
\]
\[
3x = 183
\]
\[
x = \frac{183}{3}
\]
\[
x = 61
\]

Thus, the equation for the two adjacent angles is:

\[
(2x + 3) + (x - 6) = 180
\]