To determine the lengths of two adjacent sides of the parallelogram, we need to follow these steps:
1. Understand the properties of a parallelogram:
- Opposite sides of a parallelogram are equal in length.
- Hence, we can set the expressions for opposite sides equal to each other and solve for [tex]\( n \)[/tex].
2. Given expressions for the lengths of two opposite sides:
- One side length: [tex]\( 5n - 6 \)[/tex] cm
- Opposite side length: [tex]\( 3n - 2 \)[/tex] cm
3. Set the expressions for these sides equal to solve for [tex]\( n \)[/tex]:
[tex]\[
5n - 6 = 3n - 2
\][/tex]
4. Solve the equation for [tex]\( n \)[/tex]:
- Subtract [tex]\( 3n \)[/tex] from both sides:
[tex]\[
2n - 6 = -2
\][/tex]
- Add 6 to both sides:
[tex]\[
2n = 4
\][/tex]
- Divide both sides by 2:
[tex]\[
n = 2
\][/tex]
5. Use the value of [tex]\( n \)[/tex] to calculate the lengths of the sides:
- Substitute [tex]\( n = 2 \)[/tex] into each expression:
[tex]\[
\text{Side 1: } 5n - 6 = 5(2) - 6 = 10 - 6 = 4 \, \text{cm}
\][/tex]
[tex]\[
\text{Side 3: } 2n + 3 = 2(2) + 3 = 4 + 3 = 7 \, \text{cm}
\][/tex]
6. Verify the length of the other pair of opposite sides:
- We already know that [tex]\( 3n - 2 \)[/tex] should be equal to [tex]\( 4 \, \text{cm} \)[/tex]:
[tex]\[
\text{Side 2: } 3n - 2 = 3(2) - 2 = 6 - 2 = 4 \, \text{cm}
\][/tex]
So the lengths of the two adjacent sides of the parallelogram are [tex]\(4\)[/tex] cm and [tex]\(7\)[/tex] cm.
Hence, the correct pair of adjacent sides is:
[tex]\[ \boxed{4 \text{ cm and } 7 \text{ cm}} \][/tex]
