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# 2.11: Rhombus Properties

Difficulty Level: At Grade Created by: CK-12

## Learning Objectives

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

## Perpendicular Diagonals in Rhombi

Rhombi (plural of rhombus) are equilateral.

• All four sides of a rhombus are congruent.
• Also, a square is a special kind of rhombus and shares all of the properties of a rhombus.

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.

A rhombus has four _____________________________ sides.

The diagonals of a rhombus are the same _____________________ and meet at a ________ angle, meaning they are perpendicular.

Theorem for Rhombus Diagonals

The diagonals of a rhombus are perpendicular bisectors of each other.

## 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.

Theorem for Rhombus Diagonals

The diagonals of a rhombus bisect the interior angles.

The diagonals of a rhombus are _____________________________ bisectors of each other.

The diagonals of a rhombus also ___________________________ the interior angles.

1. Fill in the blank: A rhombus is a parallelogram with congruent ________________________.

2. Label the right angles in the picture below:

3. What is the measure of angle \begin{align*}ABC\end{align*} in the rhombus below?

## Biconditional Statements

A biconditional statement is a conditional statement that also has a true converse.

For example, a true biconditional statement is, “If a quadrilateral is a square then it has exactly four congruent sides and four congruent angles.” This statement is true, as is its converse: “If a quadrilateral has exactly four congruent sides and four congruent angles, then that quadrilateral is a square.”

A biconditional statement is a true if-then statement whose _________________________ is also true.

Remember...

A conditional statement is an “if-then” statement.

A converse is a statement in which the hypothesis and conclusion are reversed.

Sometimes converses are true and sometimes they are not.

When a conditional statement can be written as a biconditional, then we use the term “if and only if.” In the previous example, we could say: “A quadrilateral is a square if and only if it has four congruent sides and four congruent angles.”

Example 1

Which of the following is a true biconditional statement?

A. A polygon is a square if and only if it has four right angles.

B. A polygon is a rhombus if and only if its diagonals are perpendicular bisectors.

C. A polygon is a parallelogram if and only if its diagonals bisect the interior angles.

D. A polygon is a rectangle if and only if its diagonals bisect each other.

Examine each of the statements to see if it is true:

A. A polygon is a square if and only if it has four right angles.

• It is true that if a polygon is a square, it has four right angles. However, the converse statement is not necessarily true. A rectangle also has four right angles, and a rectangle is not necessarily a square. Providing an example that shows something is not true is called a counterexample.

B. A polygon is a rhombus if and only if its diagonals are perpendicular bisectors.

• The second statement seems correct. It is true that rhombi have diagonals that are perpendicular bisectors. The same is also true in converse—if a figure has perpendicular bisectors as diagonals, it is a rhombus.

C. A polygon is a parallelogram if and only if its diagonals bisect the interior angles.

• The third statement is not necessarily true. Not all parallelograms have diagonals that bisect the interior angles. This is true only of rhombi, not all parallelograms.

D. A polygon is a rectangle if and only if its diagonals bisect each other.

• This is not necessarily true. The diagonals in a rectangle do bisect each other, but parallelograms that are not rectangles also have bisecting diagonals. Choice D is not correct.

So, after analyzing each statement carefully, only B is true. Choice B is the correct answer.

1. Write the following biconditional statement as an “if and only if” statement:

The sun is the star at the center of our solar system. _______________________________________________________ if and only if

_________________________________________________________________.

2. Is the following statement a true biconditional statement? If not, provide a counterexample.

A polygon is a quadrilateral if and only if it has four sides.

## Graphic Organizer for Lesson 8

Logic Statements
Type of Statement Description Example
Conditional Statement
• “________________” statement
If a shape is a polygon, then it has straight sides.
Converse
• The hypothesis and conclusion are
If a shape has straight sides, then it is a polygon.
__________________________
• Not always true
• Can be disproven with a
____________________
Biconditional Statement
• Both the statement and its
If a polygon has three sides, then it is a triangle.
____________________ are true.
• Statement is true.
• Good definitions are biconditional.
• Converse (if it’s a triangle, then it’s a polygon with three sides) is also true.

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