Final answer:
The measure of angle ACD is 30°, based on the properties of an isosceles triangle and the 30-60-90 right triangle formed by the median.
Step-by-step explanation:
The measure of angle ACD in an isosceles triangle with base angles of 30° is 30°. Since triangle ABC is isosceles with AC=BC, and the base angles are equal, angles CAB and CBA are both 30°. When we draw median CD to the base AB, it not only bisects AB but also forms two 30-60-90 right triangles, ACD and BCD.
In every 30-60-90 triangle, the angles are always in the ratio of 1:2:3, meaning 30°:60°:90°. Since angle ACD is opposite the shortest side AD (which is half of AB), it is the smallest angle, making it 30°.
First, let's analyze the given information. We know that triangle ABC is an isosceles triangle with AC = BC. We also know that CD is the median to the base.
Since AC = BC, the base angles of triangle ABC are equal. Let's call this angle x. So, we have x + x + 30° = 180° (sum of angles in a triangle). Simplifying, we get 2x + 30° = 180°. Subtracting 30° from both sides, we get 2x = 150°. Divide by 2, and we find x = 75°.
Now, angle ACD is equal to the base angle x. Therefore, angle ACD = 75°. So, the correct answer is (d) 120°.