Two figures are congruent by definition if all of their corresponding sides are congruent and all of their corresponding angles are congruent.

By definition, a rhombus has four sides of equal length.  If we let \(\displaystyle s_{1}\) be the common sidelength of Rhombus \(\displaystyle ABCD\) and \(\displaystyle s_{2}\) be the common sidelength of Rhombus \(\displaystyle EFGH\), then, since \(\displaystyle AB = EF\), it follows that \(\displaystyle s_{1} = s_{2}\), so corresponding sides are congruent. However, no information is given about their angle measures. Therefore, it cannot be determined whether or not the two rhombuses are congruent.

Two figures are similar by definition if all of their corresponding sides are proportional and all of their corresponding angles are congruent.

By definition, a rhombus has four sides that are congruent. If we let \(\displaystyle s_{1}\) be the common sidelength of Rhombus \(\displaystyle ABCD\) and \(\displaystyle s_{2}\) be the common sidelength of Rhombus \(\displaystyle EFGH\), it can easily be seen that the ratio of the length of each side of the former to that of the latter is the same ratio, namely, \(\displaystyle \frac{s_{1}}{s_{2}}\).

Also, a rhombus being a parallelogram, its opposite angles are congruent, and its consecutive angles are supplementary. Therefore, since \(\displaystyle m \angle A = 60 ^{\circ }\), it follows that \(\displaystyle m\angle C = 60 ^{\circ }\), and \(\displaystyle m \angle B = m \angle D = 180 ^{\circ } - 60^{\circ } = 120 ^{\circ }\). By a similar argument, \(\displaystyle m \angle H = m \angle F = 120 ^{\circ }\) and \(\displaystyle m \angle E = m \angle G= 180 ^{\circ } - 120 ^{\circ } = 60^{\circ }\). Therefore, 

\(\displaystyle \angle A \cong \angle E\)

\(\displaystyle \angle B \cong \angle F\)

\(\displaystyle \angle C \cong \angle F\)

\(\displaystyle \angle D \cong \angle H\)

Since all corresponding sides are proportional and all corresponding angles are congruent, it holds that Rhombus \(\displaystyle ABCD \sim\) Rhombus \(\displaystyle EFGH\).

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Thus, if a rhombus has two interior angles of \(\displaystyle 56\) degrees, there must also be two angles that equal: 

\(\displaystyle 180-56=124\)


Check:

\(\displaystyle 56+56+124+124=360\)

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Since \(\displaystyle \measuredangle C=79^\circ\) then, \(\displaystyle \measuredangle A=79^\circ\)

\(\displaystyle 79+79=158^\circ\)

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees. The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Since \(\displaystyle \measuredangle C=79^\circ\),  \(\displaystyle \measuredangle B =180-79=101^\circ\)

And \(\displaystyle \measuredangle B=\measuredangle D\) 

So, \(\displaystyle 101+101=202^\circ\)

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Since angle \(\displaystyle Y\) is adjacent to angle \(\displaystyle X\), they must have a sum of \(\displaystyle 180\) degrees. 

The solution is:

\(\displaystyle 180-63=117^\circ\)

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Angles \(\displaystyle X\) and \(\displaystyle Z\) are opposite interior angles, so they must have equivalent measurements.

The sum is:

\(\displaystyle 63+63=126^\circ\)

The four interior angles in any rhombus must have a sum of \(\displaystyle 360\) degrees.

The opposite interior angles must be equivalent, and the adjacent angles have a sum of \(\displaystyle 180\) degrees. 

Since, both angles \(\displaystyle W\) and \(\displaystyle Y\) are adjacent to angle \(\displaystyle X\)--find the measurement of one of these two angles by: \(\displaystyle 180-63=117\).

Angle \(\displaystyle W\) and angle \(\displaystyle Y\) must each equal \(\displaystyle 117\) degrees. So the sum of angles \(\displaystyle W\) and \(\displaystyle Y=117+117=234\) degrees. 

A rhombus is defined to be a parallelogram with four congruent sides; there is no restriction as to the measures of the angles. Therefore, a rhombus can have angles of any measure. The correct choice is "false".

One characteristic of a rhombus is that its diagonals are perpendicular. It follows that \(\displaystyle \angle AXB\) must be a right angle.