Question

Find the angles of the rhombus if the ratio of the angles formed by the diagonals and the sides is 4:5. I will give the brainiest answer to the one who gets it correct. PLEASE HELP There are 4 angles

Answer 1

In any rhombus, the diagonals are bisectors of the angles. For example, in rhombus attached, angle BAC is half of BAD, and so on.

So, since they are all in the same proportion, the angles of the rhombus are also in a 4:5 proportion.

We also know that the sum of the interior angles of a rhombus is 360, so we have

$\begin{cases}\alpha = \dfrac{4}{5}\beta}\\2\alpha+2\beta=360\end{cases}$

We can use the expression for $\alpha$ in the first equation to rewrite the second as

$2\alpha+2\beta=360 \iff 2\cdot\dfrac{4}{5}\beta}+2\beta = 360 \iff\left(\dfrac{8}{5}+2\right)\beta = 360 \iff \beta = 100$

And we can conclude that

$\alpha = \dfrac{4}{5}\beta=\dfrac{4}{5}\cdot 100 = 80$