Question

clara writes the following proof for the theorem: if the diagonals of a quadrilateral bisect each other, the quadrilateral is a parallelogram:claras proof1. for triangles aob and cod, angle 1 is equal to angle 2, as they are vertical angles.2. ao = oc and bo = od because it is given that diagonals bisect each other3. the triangles aob and cod are congruent by sas postulate.4. similarly, triangles aod and cob are congruent5. by ____, angle abd is equal to angle cdb and angle adb is equal to angle cbd6. as the alternate interior angles are congruent, the opposite sides of quadrilateral abcd are parallel7. therefore, abcd is a parallelogramwhich is the missing phrase in claras proof?cpctcproperty of parallelogramstransitive propertyvertical angles theorem

Explanation:

CPCTC stands for "Corresponding Parts of Congruent Triangles are Congruent". Since triangles AOB and COD are congruent (from step 3) and triangles AOD and COB are congruent (from step 4), when we want to state that angle ABD is equal to angle CDB and angle ADB is equal to angle CBD (which are corresponding parts of these congruent triangles), we use the CPCTC principle. The property of parallelograms can't be used here as we are in the process of proving the quadrilateral is a parallelogram. The transitive property is about \(a = b\) and \(b = c\) then \(a = c\), which is not the case here. The vertical angles theorem was already used in step 1.

Answer:

CPCTC