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Rhombus Properties - Problem 1
A rhombus, like a parallelogram, has two pairs of congruent, parallel sides. However, unlike a parallelogram, all sides of a rhombus are congruent.
The diagonals of the parallelogram bisect the angles that they connect. Therefore, the diagonal divides the vertex angles into two congruent pieces, so the angles formed have the same measure. Additionally, opposite angles of a rhombus are congruent, so they have the same measure, which is [360° - 2*(the measure of the other opposite angle)]/2.
We can use properties of a rhombus to find missing angles. So if we start with angle X, well it looks like angle X might be congruent to 30 degrees. If we look at our diagram here we see that this angle is bisected by that diagonal. So these two angles must be congruent, which means X must also be 30 degrees. So I’m going to write in 30 degrees here.
Now how are we going to find Y when we only know one angle in this triangle? Well, if you look at this we have a transversal which I’ll extend right here to give you a better visual of these two parallel lines which I guess I could extend as well and we have alternate interior angles. So these two angles must be congruent to each other, which means if they’re congruent this has to be 30 degrees and the sum of these three angles, Y plus 30 plus 30 must be 180 degrees.
So if you’re solving this, Y plus 60 is equal to 180, subtract 60 and Y is 120 degrees. The key thing here was realizing that this diagonal bisects that vertex’s angle and that alternate interior angles are formed by that diagonal creating two congruent angles.