Under what circumstances will ∠ADE in rhombus ABCD be obtuse?
Hint
Two forces are at work here: the fact that diagonals of a rhombus are perpendicular and the angle sum theorem for triangles.
Answer
Never
Example 2
A rhombus is a parallelogram by definition. Prove that the converse statement ("A parallelogram is a rhombus") is false.
Hint
Another way of phrasing the conclusion is, "A parallelogram is not a rhombus." Notice the emphasized "not."
Answer
Let WXYZ be a parallelogram. In this parallelogram, sides ZW and XY are congruent and parallel, and sides WX and YZ are congruent and parallel. That's it. Nowhere does it say that WX must be congruent to XY or ZW to YZ. Since that's what defines a rhombus, we've successfully proved that WXYZ can be a parallelogram and not a rhombus.
Example 3
Given that ABCD is a rhombus, what is the value of x?
Hint
Quadrilaterals might have more to do with triangles than you think. Especially in this case.
Answer
x = 5
Example 4
The diagonals of rhombus FRED are 12 ft and 16 ft. What is FRED's perimeter?
Hint
Why are a rhombus's diagonals so special, again?
Answer
40 ft
Example 5
Find the lengths of the diagonals of rhombus KYLE, KL and YE, given that the measure of ∠KYL is 20° and that KYLE's perimeter is 52 meters.
Hint
Betcha didn't think that trigonometry would be so crucial to rhombi, did you? (Please don't hate us.)
