Question

Given a rhombus in the picture below,what is the measurement of angel 2?

Answer 1

A rhombus has:

- 2 opposite angles which are congruent and

- the four angles add up 360 degrees.

The first statement means that

$\angle1+\angle2=\angle3+\angle4\ldots(A)$

The second statement means that

$(\angle1+\angle2)+(\angle3+\angle4)+130+130=360$

By rewritten this equation, we have

$\begin{gathered} (\angle1+\angle2)+(\angle3+\angle4)+260=360 \\ (\angle1+\angle2)+(\angle3+\angle4)=360-260 \\ (\angle1+\angle2)+(\angle3+\angle4)=100\ldots(B) \end{gathered}$

By substituying equation A into B, we have

$\begin{gathered} (\angle3+\angle4)+(\angle3+\angle4)=100 \\ 2(\angle3+\angle4)=100 \\ \angle3+\angle4=\frac{100}{2} \\ \angle3+\angle4=50\ldots(C) \end{gathered}$

A particular property of rhombus is that

$\begin{gathered} \angle1=\angle2 \\ \text{and} \\ \angle3=\angle4 \end{gathered}$

By substituying the last equality into equation C, we have

$\begin{gathered} \angle4+\angle4=50 \\ 2\angle4=50 \\ \angle4=\frac{50}{2} \\ \angle4=25 \end{gathered}$

Therefore, we have

$\angle3=25$

And finally, we can see that, necesarilly,

$\angle1=\angle2=\angle3=\angle4=25$