The area of a parallelogram is $3$ , and the lengths of its sides are $1$ . $1$ and $4$ . $8$ . Determine, to the nearest tenth of a degree, the measure of the obtuse angle of the parallelogram. $\newline$ Answer:

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Q. The area of a parallelogram is

3

, and the lengths of its sides are

1

1

and

4

8

. Determine, to the nearest tenth of a degree, the measure of the obtuse angle of the parallelogram.

\newline

Answer:

Identify Parallelogram Area Formula: To find the measure of the obtuse angle, we need to use the formula for the area of a parallelogram, which is $A = \text{base} \times \text{height}$ . Here, we can consider the sides of lengths $1.1$ and $4.8$ as the base and the corresponding height, respectively. However, since we are looking for the obtuse angle, we need to find the height that corresponds to the side of length $4.8$ as the base.
Calculate Height Using Area Formula: The area of the parallelogram is given as $3$ . We can use the formula $A = \text{base} \times \text{height}$ , where the base is $4.8$ . Let's denote the height as $h$ . So, we have $3 = 4.8 \times h$ .
Find Sine of Acute Angle: To find the height $h$ , we divide both sides of the equation by $4.8$ . So, $h = \frac{3}{4.8}$ .
Calculate Acute Angle: Calculating the height $h$ gives us $h = 0.625$ .
Find Obtuse Angle: Now that we have the height, we can find the sine of the acute angle $\theta$ between the base and the height using the formula $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, the opposite side is the height $h$ , and the hypotenuse is the side of length $1.1$ . So, $\sin(\theta) = \frac{0.625}{1.1}$ .
Find Obtuse Angle: Now that we have the height, we can find the sine of the acute angle ( $\theta$ ) between the base and the height using the formula $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, the opposite side is the height $h$ , and the hypotenuse is the side of length $1.1$ . So, $\sin(\theta) = \frac{0.625}{1.1}$ .Calculating $\sin(\theta)$ gives us $\sin(\theta) \approx 0.56818$ .
Find Obtuse Angle: Now that we have the height, we can find the sine of the acute angle $\theta$ between the base and the height using the formula $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, the opposite side is the height $h$ , and the hypotenuse is the side of length $1.1$ . So, $\sin(\theta) = \frac{0.625}{1.1}$ .Calculating $\sin(\theta)$ gives us $\sin(\theta) \approx 0.56818$ .To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.56818$ . $\theta = \arcsin(0.56818)$ .
Find Obtuse Angle: Now that we have the height, we can find the sine of the acute angle $\theta$ between the base and the height using the formula $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, the opposite side is the height $h$ , and the hypotenuse is the side of length $1.1$ . So, $\sin(\theta) = \frac{0.625}{1.1}$ . Calculating $\sin(\theta)$ gives us $\sin(\theta) \approx 0.56818$ . To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.56818$ . $\theta = \arcsin(0.56818)$ . Calculating $\theta$ gives us $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $1$ . Since this is the acute angle and we are looking for the obtuse angle, we subtract this from $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $2$ to find the obtuse angle. The obtuse angle $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $3$ .
Find Obtuse Angle: Now that we have the height, we can find the sine of the acute angle ( $\theta$ ) between the base and the height using the formula $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ . In this case, the opposite side is the height $h$ , and the hypotenuse is the side of length $1.1$ . So, $\sin(\theta) = \frac{0.625}{1.1}$ .Calculating $\sin(\theta)$ gives us $\sin(\theta) \approx 0.56818$ .To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.56818$ . $\theta = \arcsin(0.56818)$ .Calculating $\theta$ gives us $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $1$ . Since this is the acute angle and we are looking for the obtuse angle, we subtract this from $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $2$ to find the obtuse angle. The obtuse angle $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $3$ .Calculating $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $4$ gives us $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $5$ . This is the measure of the obtuse angle of the parallelogram, to the nearest tenth of a degree.