ABCD is a square and ΔMAB is an equilateral triangle. MC and MD are joined. What is the degree measure of ∠MDC ?

75°

ABCD is a square, = AB = BC = CD = DA

MAB is an equilateral triangle, = MA = AB = BM

square ABCD and equilateral triangle MAB have a common side AB. 

= AB = BC = CD = DA = MA = MB

In △MAD,

= MA = DA

∠ADM = ∠AMD = x

With ∠DAM

= ∠DAM = ∠MAB + ∠DAB

= ∠DAM = 90° + 60°

= ∠DAM = 150°

Sum of angles of triangle MAD = 180°

= ∠DAM + ∠AMD + ∠ADM = 180°

= 150° + x + x = 180°

= 2x = 180° - 150° 

= x = 15° 

Hence, ∠ADM = ∠BCM = 15° 

Now for ∠MDC, = ∠MDC = ∠ADC - ∠ADM

= ∠MDC = 90° - 15° = 75°