The area of a parallelogram is $22$ , and the lengths of its sides are $8$ . $8$ and $4$ . $6$ . 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

22

, and the lengths of its sides are

8

8

and

4

6

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

\newline

Answer:

Area Formula Explanation: The area of a parallelogram is given by the formula $A = \text{base} \times \text{height}$ , where the base is the length of one side and the height is the perpendicular distance from the base to the opposite side.
Given Values: We are given the area $A = 22$ and the lengths of the sides are $8.8$ and $4.6$ . We can assume that one of these sides is the base. Without loss of generality, let's take the side of length $8.8$ as the base.
Calculate Height: To find the height $h$ , we use the area formula: $A = \text{base} \times \text{height}$ . Plugging in the values we have, we get $22 = 8.8 \times h$ .
Calculate Sine of Angle: Solving for $h$ , we divide both sides of the equation by $8.8$ to get $h = \frac{22}{8.8}$ .
Calculate Acute Angle: Calculating the height $h$ , we get $h = 2.5$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ . Calculating $\sin(\theta)$ , we get $\sin(\theta) \approx 0.5435$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ . Calculating $\sin(\theta)$ , we get $\sin(\theta) \approx 0.5435$ . To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.5435$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ . Calculating $\sin(\theta)$ , we get $\sin(\theta) \approx 0.5435$ . To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.5435$ . Calculating $\theta$ , we get $\theta \approx \arcsin(0.5435) \approx 32.9°$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ . Calculating $\sin(\theta)$ , we get $\sin(\theta) \approx 0.5435$ . To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.5435$ . Calculating $\theta$ , we get $\theta \approx \arcsin(0.5435) \approx 32.9^\circ$ . Since we are looking for the measure of the obtuse angle, and we know that the sum of the angles in a parallelogram is $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $0$ , with opposite angles being equal, the obtuse angle is $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $1$ .
Calculate 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 $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ , where the opposite side is the height and the hypotenuse is the side length of $4.6$ . Plugging in the values, we get $\sin(\theta) = \frac{2.5}{4.6}$ . Calculating $\sin(\theta)$ , we get $\sin(\theta) \approx 0.5435$ . To find the measure of the acute angle $\theta$ , we take the inverse sine (arcsin) of $0.5435$ . Calculating $\theta$ , we get $\theta \approx \arcsin(0.5435) \approx 32.9^\circ$ . Since we are looking for the measure of the obtuse angle, and we know that the sum of the angles in a parallelogram is $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $0$ , with opposite angles being equal, the obtuse angle is $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $1$ . Calculating the obtuse angle, we get $\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}$ $2$ .