if triangle abc is an equilateral triangle then calculate and mention the ratio between Angle ABC and Angle CAD formed in triangle ACD which formed by joining vertex A and point D in such a way that AC=CD

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.

If triangle ABC is an equilateral triangle, then calculate and mention the ratio between Angle ABC and Angle CAD formed in triangle ACD, which is formed by joining vertex A and point D in such a way that AC = CD.

Calculate the ratio between Angle ABC and Angle CAD in a triangle where ABC is equilateral, and D is on the extended side such that AC = CD. This problem involves geometric concepts such as equilateral and isosceles triangles, angle ratios, and theorems like the Exterior Angle Theorem. Suitable for Grades 9-11.

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