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Angles in a parallelogram

The angles in a parallelogram follow two important principles. The angles opposite are equal to one another and the sum of the two angles next to each other measures 180°180 \degree, or π\pi in radian. This means if one angle is given the other angle can be computed with the following formulaα=180°β(=πβ)β=180°α(=πα). \begin{aligned} \alpha &= 180 \degree - \beta \left( = \pi - \beta \right) \\[1em] \beta &= 180 \degree - \alpha \left( = \pi - \alpha \right). \end{aligned} On the other hand if the other angle is not given then the angle can be computed using the length of the belonging side and the length of the other height known in the parallelogram. So we have the following formulaα=sin1(hba)β=sin1(hab). \begin{aligned} \alpha &= \sin^{-1} \left( \frac{h_b}{a} \right) \\[2em] \beta &= \sin^{-1} \left( \frac{h_a}{b} \right). \end{aligned} Important here is that the quotient of other height and belonging side is smaller or equal to 11, i.e. that other height is smaller or equal to that side (habh_a \leq b und hbah_b \leq a). This quotient is equal to the sine of an angle and the image of the sine is [1,1][-1,1] which is why this is so important. Otherwise the form is not a parallelogram.