CK12 - Geometry

The diagonalsof a rhombusnot only bisecteachother(becausethey are parallelograms),they do so at a
right angle.In otherwords,the diagonalsare perpendicular. This can be very helpfulwhenyou needto
measureanglesinsiderhombior squares.

Theorem:The diagonalsof a rhombusare perpendicularbisectorsof eachother

The proofof this theoremusesthe fact that the diagonalsof a parallelogrambisecteachotherand that if
two anglesare congruentand supplementary, then they are right angles.

• Given:Rhombus with diagonals and intersectingat pointA


• Prove:

Statement Reason

1. is a rhombus 1. Given


2. is a parallelogram 2. Theorem:All rhombiare parallelograms


3. 
   
   
   
   3. Definitionof a rhombus
   
   
   
   


4. 
   
   
   
   4. ReflexivePropertyof
   
   
   5. Diagonalsof a parallelogrambisecteach
      other
   
   
   
   

5.

6. SSS TriangleCongruencePostulate


7. 
   


8. Definitionof congruenttriangles(corre-
   spondingpartsof congruenttrianglesare

7.

congruent)

8. and are supplemen- 8. LinearPair Postulate
   tary


9. Congruent supplementaryangles are
   right angles


10. and are right angles


11. 
    
    
    
    10. Definitionof perpendicularlines
    
    
    
    

Rememberthat you can also showthat linesor segmentsare perpendicularby comparingtheir slopes.
Perpendicularlines haveslopesthat are oppositereciprocalsof eachother.

Example 2

Analyzethe slopeof the diagonalsin the rhombusbelow. Use slopeto demonstratethat they are perpen-
dicular.