CK12 - Geometry

Theorem:The diagonalsof a rhombusbisectthe interiorangles

Example 3

Completethe two-columnproofbelow.

• Given: is a rhombus


• Prove:

Statement Reason

1. is a rhombus 1. Given


2. 
   
   
   
   2. All sidesin a rhombusare congruent
   
   
   3. Any trianglewith two congruentsidesis
      isosceles
   
   
   
   


3. is isosceles
   4. The baseanglesin an isoscelestriangle
   are congruent

4.

5. 
   
   
   
   5. Alternateinterioranglesare congruent
   
   
   
   


6. 
   
   
   
   6. TransitiveProperty
   
   
   
   

Segment bisects. You couldwritea similarprooffor everyanglein the rhombus.Diagonals
in rhombibisectthe interiorangles.

BiconditionalStatements

Recallthat aconditionalstatementis a statementin the form “If ... then ... .” For example,if a quadrilateral
is a parallelogram,then oppositesidesare congruent.

You havelearneda numberof theoremsas conditionalstatements.Manytimesyou havealso investigated
the conversesof thesetheorems.Sometimesthe converseof a statementis true, and sometimesthe converse
are not. For example,you couldsay thatif you live in Los Angeles,you live in California. However, the
converseof this statementis not true.If you live in California,you don’tnecessarilylive in Los Angeles.

Abiconditionalstatementis a conditionalstatementthat also has a true converse.For example,a true
biconditionalstatementis, “If a quadrilateralis a squarethen it has exactlyfour congruentsidesand four
congruentangles.”This statementis true, as is its converse:“If a quadrilateralhas exactlyfour congruent
sidesand four congruentangles,then that quadrilateralis a square.”Whena conditionalstatementcan be
writtenas a biconditional,then we use the term “if and only if.” In the previousexample,we couldsay: “A
quadrilateralis a squareif and only if it has four congruentsidesand four congruentangles.”

Example 4