College Geometry

Question

Prove that an inscribed rhombus is a square. (HINT. One way to proceed is to use the fact that opposite angles of an inscribed quadrilateral are supplementary.)

Answer 1

rhombus definition is four sides all equal length connected at four vertices , no requirement on angles at vertices

I think inscribed means putting a rhombus in a circle

establishing radius from center to each vertex, four radial angles

angles must equal each other (four triangles congruent: side side side)

and angles must add to 360 degrees

4*A=360

a=90

each triangle is 90,45,45 degrees since two sides=radius

each vertex is therefore 45+45=90

now have four equal sides and vertex angles at 90 degrees = square

Answer 2

Draw the diagonal in the rhombus.

The 2 triangles on each side of the diagonal are congruent.

The vertex angles of these 2 triangles are equal and since they are supplementary they are each right angles.

The same argument using the other diagonal proves that all 4 angles of the rhombus are right angles.

Therefore, the rhombus is a square.