To determine the measure of the obtuse angle of a parallelogram given specific parameters, follow these steps:
1. Identify Given Values:
- The area of the parallelogram: [tex]\( \text{Area} = 6 \)[/tex]
- Lengths of the sides: [tex]\( \text{side1} = 2.1 \)[/tex] and [tex]\( \text{side2} = 3.5 \)[/tex]
2. Understand the Formula:
- The area of a parallelogram is calculated using the formula:
[tex]\[
\text{Area} = \text{side1} \times \text{side2} \times \sin(\theta)
\][/tex]
where [tex]\( \theta \)[/tex] is the angle between the two sides.
3. Rearrange the Formula to Solve for [tex]\( \theta \)[/tex]:
[tex]\[
\sin(\theta) = \frac{\text{Area}}{\text{side1} \times \text{side2}}
\][/tex]
4. Calculate [tex]\( \theta \)[/tex]:
[tex]\[
\sin(\theta) = \frac{6}{2.1 \times 3.5}
\][/tex]
- First, calculate the denominator: [tex]\( 2.1 \times 3.5 = 7.35 \)[/tex]
- Next, find the ratio: [tex]\( \sin(\theta) = \frac{6}{7.35} \approx 0.8163 \)[/tex]
- Then, determine the angle [tex]\( \theta \)[/tex] in radians using the inverse sine function:
[tex]\[
\theta = \sin^{-1}(0.8163) \approx 0.9550 \text{ radians}
\][/tex]
5. Convert Radians to Degrees:
- To convert the angle from radians to degrees, use the conversion factor [tex]\( 180^\circ / \pi \)[/tex]:
[tex]\[
\theta \approx 0.9550 \times \frac{180}{\pi} \approx 54.7^\circ
\][/tex]
6. Determine the Obtuse Angle:
- Since the obtuse angle is the supplement of the acute angle [tex]\( \theta \)[/tex], we subtract the acute angle from 180 degrees:
[tex]\[
\text{Obtuse Angle} = 180^\circ - 54.7^\circ \approx 125.3^\circ
\][/tex]
So, to the nearest tenth of a degree, the measure of the obtuse angle of the parallelogram is [tex]\( 125.3^\circ \)[/tex].
