<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation
Our Terms of Use (click here to view) and Privacy Policy (click here to view) have changed. By continuing to use this site, you are agreeing to our new Terms of Use and Privacy Policy.

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.

Reading Check:

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 ABC 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.

Reading Check:

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.

Image Attributions

Show Hide Details
Description
Authors:

Concept Nodes:

Grades:
8 , 9 , 10
Date Created:
Feb 23, 2012
Last Modified:
May 12, 2014
Save or share your relevant files like activites, homework and worksheet.
To add resources, you must be the owner of the section. Click Customize to make your own copy.
Reviews
Help us create better content by rating and reviewing this modality.
Loading reviews...
Please wait...
Please wait...
Image Detail
Sizes: Medium | Original