Certainly! Let's solve the problem step-by-step.
Given: Two adjacent angles of a parallelogram are [tex]\( (5x - 5^\circ) \)[/tex] and [tex]\( (10x + 35^\circ) \)[/tex].
### Step 1: Understanding the relationship between adjacent angles in a parallelogram.
In a parallelogram, the sum of two adjacent angles is [tex]\(180^\circ\)[/tex].
### Step 2: Set up the equation.
The sum of the given angles is:
[tex]\[
(5x - 5^\circ) + (10x + 35^\circ) = 180^\circ
\][/tex]
### Step 3: Simplify the equation.
Combine like terms:
[tex]\[
5x - 5^\circ + 10x + 35^\circ = 180^\circ
\][/tex]
[tex]\[
15x + 30^\circ = 180^\circ
\][/tex]
### Step 4: Solve for [tex]\(x\)[/tex].
Subtract [tex]\(30^\circ\)[/tex] from both sides:
[tex]\[
15x = 150^\circ
\][/tex]
Divide by 15:
[tex]\[
x = 10^\circ
\][/tex]
### Step 5: Calculate the angles.
Substitute [tex]\(x = 10^\circ\)[/tex] back into the expressions for the angles:
[tex]\[
\text{Angle 1} = 5x - 5^\circ = 5(10^\circ) - 5^\circ = 50^\circ - 5^\circ = 45^\circ
\][/tex]
[tex]\[
\text{Angle 2} = 10x + 35^\circ = 10(10^\circ) + 35^\circ = 100^\circ + 35^\circ = 135^\circ
\][/tex]
### Step 6: Find the ratio of these angles.
The ratio of the first angle to the second angle is:
[tex]\[
\frac{\text{Angle 1}}{\text{Angle 2}} = \frac{45^\circ}{135^\circ} = \frac{45}{135} = \frac{1}{3}
\][/tex]
Therefore, the ratio of the two adjacent angles is [tex]\( \boxed{\frac{1}{3}} \)[/tex].
