Final answers:
1. The measure of angle AMD is 127°.
2. The measure of angle ADM is 53°.
3. The measure of angle ACD is 90°.
4. The length of diagonal DM cannot be determined with the provided information.
5. The length of diagonal AC cannot be determined with the provided information.
6. Yes, the quadrilateral is a parallelogram due to the property that opposite angles of a parallelogram are congruent.
Step-by-step explanation:
1. To find angle AMD, in rhombus ABCD, where diagonals intersect at M, apply the property that opposite angles in a rhombus are equal. Since angle MAB is 53° and the diagonals of a rhombus bisect each other at right angles, angle MAB and angle MAD are congruent. Hence, angle MAD is also 53°. Therefore, angle AMD = 180° - angle MAD = 180° - 53° = 127°.
2. Angle ADM in a rhombus is equal to angle MAD as diagonals of a rhombus bisect each other. Given MB = 16 and AM = 12, this signifies that triangle AMB is a right-angled triangle by applying the Pythagorean theorem: AB^2 = AM^2 + MB^2. Substituting the values, AB^2 = 12^2 + 16^2 = 144 + 256 = 400. Hence, AB = √400 = 20. As diagonals of a rhombus bisect each other at right angles, angle ADM = angle MAD = angle MAB = 53°.
3. In a rhombus, opposite angles are equal. Angle ACD is opposite to angle AMB. Given angle AMB = 90°, then angle ACD = 90°.
4. & 5. Without further information or additional side lengths provided, the lengths of diagonals DM and AC cannot be computed as they depend on the rhombus' side lengths.
6. Yes, a quadrilateral with four congruent angles is a parallelogram. In a parallelogram, opposite angles are equal. If all four angles are congruent, then opposite angles are also equal, satisfying the condition for a parallelogram. Therefore, the quadrilateral is a parallelogram.