$9$ . The measures of two sides of a parallelogram are $28 \mathrm{in}$ . and $42$ in. If the length of the longer diagonal is $58$ in., what are the measures of the angles at the vertices? Round to the nearest tenth of a degree.

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9

. The measures of two sides of a parallelogram are

28 \mathrm{in}

. and

42

in. If the length of the longer diagonal is

58

in., what are the measures of the angles at the vertices? Round to the nearest tenth of a degree.

Use Law of Cosines: To find the angles, we can use the law of cosines in one of the triangles formed by the diagonal. $\newline$ Let's calculate the angle opposite the diagonal using the law of cosines: $c^2 = a^2 + b^2 - 2ab\cos(C)$ , where $c$ is the diagonal, and $a$ and $b$ are the sides of the parallelogram.
Calculate Diagonal Angle: Plug in the values: $58^2 = 28^2 + 42^2 - 2 \times 28 \times 42 \cos(C)$ .
Solve for Cosine: Calculate the squares: $3364 = 784 + 1764 - 2352\cos(C)$ .
Find Angle C: Combine like terms: $3364 = 2548 - 2352\cos(C)$ .
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ .
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ .Calculate the difference: $816 = -2352\cos(C)$ .
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ .Calculate the difference: $816 = -2352\cos(C)$ .Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}.$
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ . Calculate the difference: $816 = -2352\cos(C)$ . Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}$ . Calculate the division: $\cos(C) = -0.347$ .
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ .Calculate the difference: $816 = -2352\cos(C)$ .Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}$ .Calculate the division: $\cos(C) = -0.347$ .Use the inverse cosine to find the angle $C$ : $C = \cos^{-1}(-0.347)$ .
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ . Calculate the difference: $816 = -2352\cos(C)$ . Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}$ . Calculate the division: $\cos(C) = -0.347$ . Use the inverse cosine to find the angle $C$ : $C = \cos^{-1}(-0.347)$ . Calculate the angle: $C \approx 110.3$ degrees.
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ .Calculate the difference: $816 = -2352\cos(C)$ .Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}$ .Calculate the division: $\cos(C) = -0.347$ .Use the inverse cosine to find the angle $C$ : $C = \cos^{-1}(-0.347)$ .Calculate the angle: $C \approx 110.3$ degrees.Since the angles at opposite vertices of a parallelogram are equal, the other angle at the opposite vertex is also $3364 - 2548 = -2352\cos(C)$ $0$ degrees. $\newline$ The adjacent angles are supplementary in a parallelogram, so we subtract from $3364 - 2548 = -2352\cos(C)$ $1$ degrees to find the other two angles: $3364 - 2548 = -2352\cos(C)$ $2$ degrees.
Calculate Other Angles: Subtract $2548$ from both sides: $3364 - 2548 = -2352\cos(C)$ . Calculate the difference: $816 = -2352\cos(C)$ . Divide by $-2352$ to solve for $\cos(C)$ : $\cos(C) = \frac{816}{-2352}$ . Calculate the division: $\cos(C) = -0.347$ . Use the inverse cosine to find the angle $C$ : $C = \cos^{-1}(-0.347)$ . Calculate the angle: $C \approx 110.3$ degrees. Since the angles at opposite vertices of a parallelogram are equal, the other angle at the opposite vertex is also $3364 - 2548 = -2352\cos(C)$ $0$ degrees. The adjacent angles are supplementary in a parallelogram, so we subtract from $3364 - 2548 = -2352\cos(C)$ $1$ degrees to find the other two angles: $3364 - 2548 = -2352\cos(C)$ $2$ degrees. The measures of the angles at the vertices are approximately $3364 - 2548 = -2352\cos(C)$ $0$ degrees and $3364 - 2548 = -2352\cos(C)$ $4$ degrees.