given:
δabc, m∠c = 90º, m∠b = 30º
cm - angle bisector

find: m∠amc

75°

Step-by-step explanation:

1. Consider right triangle ABC. In this triangle

m∠C = 90º and CM - angle bisector.

If CM is angle C bisector, then

m∠ACM = m∠MCB = 45º

2. Consider triangle CMB. The sum of the measures of all interior angles of the triangle is always 180°, then

m∠MCB + m∠CBM + m∠BMC = 180°,

m∠BMC = 180° - 45° - 30°

m∠BMC = 105°

3. Angles AMC and BMC are supplementary angles, so

m∠AMC + m∠BMC = 180°

m∠AMC = 180° - 105°

m∠AMC = 75°