Problem 695: Triangle, Angles, Auxiliary Construction, Congruence, Mind Map, Polya. Level: High School, College, Math Ed.

The figure shows a quadrilateral ABCD with angle ABD = 54 degrees, angle DBC = 63 degrees, angle ACB = 38 degrees, and angle ACD = 27 degrees, Find the measure of the angle CAD.
 

 
 

Geometry problem solving is one of the most challenging skills for students to learn. When a problem requires auxiliary construction, the difficulty of the problem increases drastically, perhaps because deciding which construction to make is an ill-structured problem. By “construction,” we mean adding geometric figures (points, lines, planes) to a problem figure that wasn’t mentioned as "given."

George Pólya's 1945 book "How to Solve It, A new aspect of Mathematical Method", is a book describing methods of problem solving. It suggests the following steps when solving a mathematical problem: (1) First, you have to understand the problem. (2) After understanding, then make a plan. (3) Carry out the plan. (4) Look back on your work. How could it be better?