Skill: Identify properties of rhombi

A rhombus is a quadrilateral with four congruent sides. Because every rhombus is also a parallelogram, it has these properties:

• All four sides have equal length.
• Opposite sides are parallel and congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary, so they add to $180^\circ$.
• The diagonals bisect each other at right angles.
• Each diagonal bisects a pair of opposite angles.

A rhombus does not have to have right angles. A square is a special rhombus because it has four equal sides and four right angles.

Example: Rhombus $ABCD$ has $AB=9$ units, $\angle A=70^\circ$, and diagonal $AC=12$ units. Diagonals $AC$ and $BD$ intersect at $E$. Find the other side lengths, the other angles, and $AE$.

Step 1: Use the equal-side property.

All four sides of a rhombus are congruent:

$$AB=BC=CD=DA$$

Since $AB=9$ units,

$$BC=CD=DA=9\text{ units}$$

Step 2: Find the opposite angle.

Opposite angles in a rhombus are congruent. Therefore,

$$\angle C=\angle A=70^\circ$$

Step 3: Find the consecutive angles.

Consecutive angles are supplementary, so they add to $180^\circ$:

$$\angle B=180^\circ-70^\circ=110^\circ$$

The opposite angle is equal to $\angle B$, so

$$\angle D=110^\circ$$

Thus, the angles are

$$\angle A=\angle C=70^\circ,\qquad \angle B=\angle D=110^\circ$$

Step 4: Use the diagonal property.

The diagonals of a rhombus bisect each other, so $E$ is the midpoint of $AC$. Therefore,

$$AE=EC=\frac{AC}{2}=\frac{12}{2}=6\text{ units}$$

So the results are:

• Each side is $9$ units.
• $\angle A$ and $\angle C$ measure $70^\circ$.
• $\angle B$ and $\angle D$ measure $110^\circ$.
• $AE=EC=6$ units.
• The diagonals meet at a right angle.

Remember: the diagonals of a rhombus are perpendicular, but they do not have to be the same length.