Rhombus

A rhombus is a parallelogram with four congruent sides. The plural of rhombus is rhombi . (I love that word.)

The formula for the area of a rhombus is the same as the formula for a parallelogram:

$A = b h$ ,

where $b$ is the length of a base and $h$ is the height.

$Area = 5 \cdot 4 = 20 {cm}^{2}$ .

If a parallelogram is a rhombus, then its diagonals are perpendicular.

For Example: If $P Q R S$ is a rhombus, then $\bar{P R} ⊥ \bar{Q S}$ .

If a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles.

For Example: If $P Q R S$ is a rhombus, then

$\begin{array}{l} ∠ 1 ≅ ∠ 2, ∠ 3 ≅ ∠ 4, ∠ 5 ≅ ∠ 6, and ∠ 7 ≅ ∠ 8 \end{array}$ .

Example 1:

In rhombus $A B C D$ , $m ∠ A B C = 2 x - 7 and m ∠ B C D = 2 x + 3$ . Find $m ∠ D A B .$

In a rhombus, consecutive interior angles are supplementary. So,

$m ∠ A B C + m ∠ B C D = 180 °$ .

Substitute the measures and solve for $x$ .

$\begin{array}{l} 2 x - 7 + 2 x + 3 = 180 ° \\ 4 x = 184 \\ x = 46 \end{array}$

Use the value of $x$ to find $m ∠ D A B .$

Each pair of opposite angles of a rhombus is congruent. So,

$m ∠ D A B = m ∠ B C D = 2 (46) + 3 = 95 °$ .

Example 2:

In rhombus $X Y Z W$ , $m ∠ W Y Z = 63 °$ . Find $m ∠ Y Z V .$

The diagonals of a rhombus are perpendicular.

$So, m ∠ Y V Z = 90 ° and Δ Y V Z is a right triangle .$

Use the Angle Sum Theorem.

$m ∠ Y Z V + m ∠ Z V Y + m ∠ V Y Z = 180 °$

Substitute the measures and solve.

$m ∠ Y Z V + 90 ° + 63 ° = 180 °$

$\begin{matrix} m ∠ Y Z V + 153 ° = 180 ° \\ m ∠ Y Z V = 27 ° \end{matrix}$