It is given that line MO = line TR and line NP = line QS, where MNOP and TQRS are parallelograms. A student has said that if those statements are true, then MNOP = TQRS. Why is this student incorrect?

Question

Aaliyah Anderson · Accepted Answer

The student is incorrect and for parallelograms to be equal, included angles between them too should be equal.

When two quadrilaterals #MNOP=TQRS#, then not only all corresponding sides are equal, all corresponding angles too are equal.

As #MNOP# and #TQRS# are parallelograms their diagonals bisect each other. And as #MO=TR# and #NP=QS#, it is apparent (see figure below) that #MX=XO=TY=YR# and #NX=XP=SY=YQ#.

And hence two sides of all the four triangles (in each of the parallelogram) are equal.

However, the angles included between the two diagonals can change (as is seen from the figure below),

and hence #MNOP!=TQRS#

Hence, the student is incorrect and for parallelograms to be equal, included angles between them too should be equal.

Violet Aldrich · Answer

undefined The student is incorrect because the statement "MNOP = TQRS" implies that parallelograms MNOP and TQRS are congruent, meaning they are identical in shape and size. However, the given conditions, line MO = line TR and line NP = line QS, only provide information about the lengths of corresponding sides.

In order for two parallelograms to be congruent (equal in the sense of geometry), besides having pairs of corresponding sides equal in length, they must also have corresponding angles that are equal. The information provided does not include any comparison of the angles of the two parallelograms. Therefore, without information on the angles, we cannot conclude that the parallelograms are congruent, even if pairs of their corresponding sides are equal in length. Parallelograms with sides of equal length can still have different angles, resulting in different shapes and sizes overall.