Suppose you know that triangle SOK is congruent to triangle COA. Explain how you could prove that quadrilateral SACK is a parallelogram.

Question

this is a confusing question I found in my geometry textbook. Please help me solve it.

Answer 1

For proving that it is a parallelogram (opposites sides parallel and congruent by definition), prove that ∠SOA ≅ ∠COK are vertical angles (Vertical Angle Theorem), AK and SC are congruent diagonals of the parallelogram.  The triangles must be either isosceles or equilateral to obtain a parallelogram figure.  Otherwise a trapezoid is drawn.  Then....

 

AC and SK are bases of each given triangle

Point O is vertex common to both triangles

AC ≅ SK

AC || SK

SO ≅ OC........given triangles are similar

SO+OC=SC...segment addition postulate

SC is a diagonal of SACK

AO ≅ OK........given triangles are similar

AO+OK=AK...segment addition postulate

AK is a diagonal of SACK

SO ≅ AO ≅ CO ≅ KO...given triangles are similar

∠SOK ≅ ∠AOC are vertical angles, given

∠SOA ≅ ∠COK are vertical angles

SA ≅ CK
SA || CK