can you find x

[@jimlocke9320]

At 5:30, the length of a side of the small square has been determined to be y(√3 - 1). Length AE = y + y(√3 - 1) = y√3. Ratio of short side to long side of ΔABE = AB/AE = y/y√3 = 1/√3. We recognize this as the ratio of short to long side of the special 30°-60°-90° right triangle and

[@juanalfaro7522]

I obtained the same answer by working on the triangle at the bottom of the 2 squares. If I label the big square ABCG and the smaller square FDEG (where F is the intersection top point and G the bottom one, between the 2 squares. Then I label the angle BEG as "d". So then x=45-d. Tan (d) = y/(y+z), where z= y(1-tan 15) ---> tan(d) = 1/(1+1-tan 15) = 1/(2 - tan 15). Then tan(x) - tan(45-d) = (tan (45) - tan (d)) / [1 + tan(45)*tan(d)] = [1 - 1/(2-tan(15))]/[1 + 1*1/(2-tan(15)] = [2-tan(15)-1]/[2-tan(15)+1] = [1-tan(15)]/[3-tan(15)]. Now tan(15) = [sqrt(6)-sqrt(2)]/[sqrt(6)+sqrt(2)] = 2- sqrt(3) --> tan(x)=[(1-(2-sqrt(3)]/[1+(2-sqrt(3))] = [sqrt(3)-1]/[sqrt(3)+1] = 2- sqrt(3) = tan(15) ---> X=15