To solve for the measures of the two adjacent angles of the parallelogram, we need to use the property of a parallelogram stating that adjacent angles are supplementary. This means that the sum of their angles is [tex]\(180^\circ\)[/tex].
The given measures of the two adjacent angles are:
[tex]\[
(2x + 3)^\circ \quad \text{and} \quad (3x + 7)^\circ
\][/tex]
Since these angles are supplementary, we can write the following equation:
[tex]\[
(2x + 3) + (3x + 7) = 180
\][/tex]
First, combine like terms:
[tex]\[
2x + 3x + 3 + 7 = 180
\][/tex]
[tex]\[
5x + 10 = 180
\][/tex]
Next, solve for [tex]\(x\)[/tex] by isolating it on one side of the equation. Start by subtracting 10 from both sides:
[tex]\[
5x + 10 - 10 = 180 - 10
\][/tex]
[tex]\[
5x = 170
\][/tex]
Then, divide both sides by 5:
[tex]\[
x = \frac{170}{5}
\][/tex]
[tex]\[
x = 34
\][/tex]
Now that we have the value of [tex]\(x\)[/tex], we can find the measures of the two angles by substituting [tex]\(x = 34\)[/tex] back into the original expressions for the angles.
First angle:
[tex]\[
2x + 3 = 2(34) + 3 = 68 + 3 = 71^\circ
\][/tex]
Second angle:
[tex]\[
3x + 7 = 3(34) + 7 = 102 + 7 = 109^\circ
\][/tex]
Therefore, the measures of the two adjacent angles are:
[tex]\[
71^\circ \text{ and } 109^\circ
\][/tex]
