Let's break down the problem and calculate the required ratio:

Given:

• Triangle ABC is an equilateral triangle.
• Point D is such that $AC = CD$, which means $D$ lies on the extension of side $AC$.

Step-by-Step Explanation:

1. Angle in the Equilateral Triangle ABC:
   In an equilateral triangle, all three angles are equal, and each angle is $60^\circ$.
   Therefore,
   $\angle ABC = 60^\circ.$

2. Triangle ACD:

   • Since $AC = CD$, triangle ACD is isosceles.
   • The base angles of an isosceles triangle are equal. Let’s call the base angles $\theta$.
   • The sum of the angles in triangle ACD must be $180^\circ$, so: $\angle ACD + 2\theta = 180^\circ.$ Since $\angle ACD$ is the exterior angle of triangle ABC, $\angle ACD = \angle ABC = 60^\circ$. Therefore: $60^\circ + 2\theta = 180^\circ,$ $2\theta = 120^\circ,$ $\theta = 60^\circ.$

   Thus, $\angle CAD = 60^\circ$.

3. Ratio of $\angle ABC$ to $\angle CAD$:
   Both $\angle ABC$ and $\angle CAD$ are $60^\circ$. Therefore, the ratio is: $\frac{\angle ABC}{\angle CAD} = \frac{60^\circ}{60^\circ} = 1.$

Final Answer:

The ratio between $\angle ABC$ and $\angle CAD$ is 1:1.

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Would you like more details or have any questions?

Here are 5 related questions to expand this topic:

1. How can we prove the angles in an equilateral triangle are always 60°?
2. What is the sum of the internal angles of any triangle?
3. Can we determine the length of side $CD$ if $AC$ is known?
4. How do the properties of an isosceles triangle help in finding unknown angles?
5. How does extending a side of a triangle affect the internal angles?

Tip: In an isosceles triangle, knowing two sides are equal simplifies angle calculations because the base angles will always be equal.