Certainly! Let's solve the problem step-by-step.
We are given that the measures of two adjacent angles of a parallelogram are in the ratio [tex]\(6:4\)[/tex]. To find the measures of these angles, we can follow these steps:
1. Understand the Property of Parallelograms:
- The sum of the measures of two adjacent angles in a parallelogram is [tex]\(180\)[/tex] degrees.
2. Set Up the Ratio Equation:
- Let's assume the measures of the two adjacent angles are represented by [tex]\(6x\)[/tex] and [tex]\(4x\)[/tex] respectively, where [tex]\(x\)[/tex] is a common multiplier.
3. Form the Equation:
- Since the sum of the adjacent angles is [tex]\(180\)[/tex] degrees, we can write the equation:
[tex]\[
6x + 4x = 180
\][/tex]
4. Simplify the Equation:
- Combine the terms involving [tex]\(x\)[/tex]:
[tex]\[
10x = 180
\][/tex]
5. Solve for [tex]\(x\)[/tex]:
- Divide both sides of the equation by [tex]\(10\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[
x = \frac{180}{10} = 18
\][/tex]
6. Find the Measures of the Angles:
- Multiply [tex]\(x\)[/tex] by the coefficients in the ratio to find the measures of the angles:
- The measure of the first angle:
[tex]\[
6x = 6 \times 18 = 108 \text{ degrees}
\][/tex]
- The measure of the second angle:
[tex]\[
4x = 4 \times 18 = 72 \text{ degrees}
\][/tex]
Therefore, the measures of the two adjacent angles of the parallelogram are [tex]\(108\)[/tex] degrees and [tex]\(72\)[/tex] degrees.
