To find the measures of all the angles in a parallelogram where the ratio of two adjacent angles is given as [tex]\( 7:17 \)[/tex], follow these steps:
1. Understand the properties of a parallelogram:
- Adjacent angles in a parallelogram add up to [tex]\(180^\circ\)[/tex].
- Opposite angles in a parallelogram are equal.
2. Set up the relationship with a variable:
- Let's represent the two adjacent angles as [tex]\(7x\)[/tex] and [tex]\(17x\)[/tex], where [tex]\(x\)[/tex] is a common multiple.
3. Formulate the equation based on the property of adjacent angles:
- According to the property of adjacent angles in a parallelogram:
[tex]\[
7x + 17x = 180^\circ
\][/tex]
4. Solve for [tex]\(x\)[/tex]:
- Combine like terms:
[tex]\[
24x = 180^\circ
\][/tex]
- Divide both sides by 24 to find [tex]\(x\)[/tex]:
[tex]\[
x = \frac{180}{24}
\][/tex]
5. Calculate the value of [tex]\(x\)[/tex]:
- [tex]\[
x = 7.5
\][/tex]
6. Determine the measures of the angles:
- Calculate the first angle:
[tex]\[
\text{First angle} = 7x = 7 \times 7.5 = 52.5^\circ
\][/tex]
- Calculate the second angle:
[tex]\[
\text{Second angle} = 17x = 17 \times 7.5 = 127.5^\circ
\][/tex]
7. Assign these measures to corresponding angles in the parallelogram:
- Since opposite angles in a parallelogram are equal, the other two angles will also be [tex]\(52.5^\circ\)[/tex] and [tex]\(127.5^\circ\)[/tex].
Therefore, the measures of all the angles in the parallelogram are:
[tex]\[
52.5^\circ, 127.5^\circ, 52.5^\circ, 127.5^\circ
\][/tex]
