In a rhombus, an altitude from the vertex of an obtuse angle bisects the opposite side. Find the measures of the angles of the rhombus.

The obtuse angles will measure 120° and the acute angles will measure 60°.  

Drawing the altitude forms a right triangle.  Let the side of the rhombus be x.  The altitude bisects the opposite side, so the base of the triangle formed would be 1/2x.  We have an expression for the side opposite this portion of the obtuse angle (cut by the altitude) and for the hypotenuse (the side length of the rhombus, x).  opposite/hypotenuse is the ratio for sine.  Our equation then looks like this:
[tex]\sin x=\frac{\frac{1}{2}x}{x} \\ \sin x=\frac{1}{2}x \div x \\ \sin x=\frac{1}{2}x \div \frac{x}{1} \\ \sin x=\frac{1x}{2} * \frac{1}{x}=\frac{1x}{2x}=\frac{1}{2}[/tex]
Now we take the inverse sine of both sides:
sin⁻¹(sin x)=sin⁻¹(1/2)
x=30

Since this portion of the triangle is 30, and the right angle is 90, the missing angle (an acute angle of the rhombus) is 180-30-90=60°.  Since the acute angles and obtuse angles of a rhombus are supplementary, the obtuse angles must be 180-60=120°.