Rhombi

Learning Objectives

  • Identify the relationship between diagonals in a rhombus.
  • Identify the relationship between diagonals and opposite angles in a rhombus.
  • Identify and explain biconditional statements.

Introduction

Now that you have a much better understanding of parallelograms, you can begin to look more carefully into certain types of parallelograms. Remember that all of the rules that apply to parallelograms still apply to rhombi. In this lesson, you’ll learn about rules specific to these shapes that are not true for all parallelograms.

Perpendicular Diagonals in Rhombi

Remember that rhombi are quadrilaterals that have four congruent sides. They don’t necessarily have right angles (like squares), but they are also parallelograms. Also, all squares are parallelograms.

The diagonals of a rhombus not only bisect each other (because they are parallelograms), they do so at a right angle. In other words, the diagonals are perpendicular. This can be very helpful when you need to measure angles inside rhombi or squares.

Theorem: The diagonals of a rhombus are perpendicular bisectors of each other

The proof of this theorem uses the fact that the diagonals of a parallelogram bisect each other and that if two angles are congruent and supplementary, then they are right angles.

  • Given: Rhombus RMBS with diagonals \overline{RB} and \overline{MS} intersecting at point A
  • Prove: \overline{RB} \perp \overline {MS}

Statement

Reason

1. RMBS is a rhombus

1. Given

2. RMBS is a parallelogram

2. Theorem: All rhombi are parallelograms

3. \overline{RM} \cong \overline {MB}

3. Definition of a rhombus

4. \overline{AM} \cong \overline {AM}

4. Reflexive Property of \cong

5. \overline{RA} \cong \overline {AB}

5. Diagonals of a parallelogram bisect each other

6. \triangle {RAM} \cong \triangle {BAM}

6. SSS Triangle Congruence Postulate

7. \angle {RAM} \cong \angle {BAM}

7. Definition of congruent triangles (corresponding parts of congruent triangles are congruent)

8. \angle {RAM} and \angle {BAM} are supplementary

8. Linear Pair Postulate

9. \angle {RAM} and \angle {BAM} are right angles

9. Congruent supplementary angles are right angles

10. \overline{RB} \perp \overline {MS}

10. Definition of perpendicular lines

Remember that you can also show that lines or segments are perpendicular by comparing their slopes. Perpendicular lines have slopes that are opposite reciprocals of each other.

Example 2

Analyze the slope of the diagonals in the rhombus below. Use slope to demonstrate that they are perpendicular.

Notice that the diagonals in this diagram have already been drawn in for you. To find the slope, find the change in y over the change in x. This is also referred to as rise over run.

Begin by finding the slope of the diagonal \overline {WY}, which goes from W(-3,2) to Y(5,-2).

\text{slope of }\overline {WY} &= \frac{y_2-y_1} {x_2-x_1} \\ &= \frac{(-2)-2} {5-(-3)} \\ &= \frac{-4} {8} \\ &= -\frac{1} {2}

Now find the slope of the diagonal \overline {ZX} from Z(0,-2) to X(2,2).

\text{slope of }\overline {ZX} &= \frac{y_2-y_1} {x_2-x_1} \\ &= \frac{2-(-2)} {2-0} \\ &= \frac{4} {2} \\ &= \frac{2} {1} \\ &=2

The slope of \overline{WY} = - \frac{1} {2} and the slope of \overline{ZX} = 2 . These two slopes are opposite reciprocals of each other, so the two segments are perpendicular.

Diagonals as Angle Bisectors

Since a rhombus is a parallelogram, opposite angles are congruent. One property unique to rhombi is that in any rhombus, the diagonals will bisect the interior angles. Here we will prove this theorem using a different method than the proof we showed above.

Theorem: The diagonals of a rhombus bisect the interior angles

Example 3

Complete the two-column proof below.

  • Given: ABCD is a rhombus
  • Prove: \angle {BDA} \cong \angle {BDC}

Statement

Reason

1. ABCD is a rhombus

1. Given

2. \overline {DC} \cong \overline {BC}

 2. All sides in a rhombus are congruent

3. \triangle {BCD} is isosceles

3. Any triangle with two congruent sides is isosceles

4. \angle {BDC} \cong \angle {DBC}

 4. The base angles in an isosceles triangle are congruent

5. \angle {BDA} \cong \angle {DBC}

 5. Alternate interior angles are congruent

6. \angle {BDA} \cong \angle {BDC}

 6. Transitive Property

Segment BD bisects \angle {ADC}. You could write a similar proof for every angle in the rhombus. Diagonals in rhombi bisect the interior angles.

Lesson Summary

In this lesson, we explored rhombi. Specifically, we have learned:

  • How to identify and prove the relationship between diagonals in a rhombus.
  • How to identify and prove the relationship between diagonals and opposite angles in a rhombus.

It is helpful to be able to identify specific properties in quadrilaterals. You will be able to use this information in many different ways.

Last modified: Monday, June 28, 2010, 4:59 PM