2.11: Rhombus Properties
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 \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 ifthen statement whose _________________________ is also true.
Remember...
A conditional statement is an “ifthen” 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
Type of Statement  Description  Example 

Conditional Statement 

If a shape is a polygon, then it has straight sides. 
Converse 

If a shape has straight sides, then it is a polygon. 
__________________________  




____________________  
Biconditional Statement 

If a polygon has three sides, then it is a triangle. 
____________________ are true. 




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