To find the ratio of the adjacent angles of a parallelogram given by [tex]\((5x - 5)^\circ\)[/tex] and [tex]\((10x + 35)^\circ\)[/tex], we follow these steps:
1. Understand the properties of a parallelogram:
- The sum of the adjacent angles in a parallelogram is [tex]\(180^\circ\)[/tex].
2. Set up the equation based on the property:
[tex]\[
(5x - 5) + (10x + 35) = 180
\][/tex]
3. Simplify the equation:
[tex]\[
5x - 5 + 10x + 35 = 180
\][/tex]
[tex]\[
15x + 30 = 180
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Subtract 30 from both sides:
[tex]\[
15x = 150
\][/tex]
- Divide both sides by 15:
[tex]\[
x = 10
\][/tex]
5. Find the actual measures of the angles:
- Substitute [tex]\(x = 10\)[/tex] back into the expressions for the angles:
[tex]\[
A = 5x - 5 = 5(10) - 5 = 50 - 5 = 45^\circ
\][/tex]
[tex]\[
B = 10x + 35 = 10(10) + 35 = 100 + 35 = 135^\circ
\][/tex]
6. Calculate the ratio of the angles:
- The ratio of angle [tex]\(A\)[/tex] to angle [tex]\(B\)[/tex] is:
[tex]\[
\frac{A}{B} = \frac{45}{135} = \frac{1}{3}
\][/tex]
7. Express the ratio in simplest form:
- Therefore, the ratio is:
[tex]\[
1:3
\][/tex]
Thus, the ratio of the two given adjacent angles in the parallelogram is [tex]\(\boxed{1:3}\)[/tex].
