https://file.aiquickdraw.com/mathbot/file/1728303426645tpdumd7b.jpg

Let's break down the information from the image:

- The problem states that Triangle **ADB** is an **equilateral triangle**.
- Line segment **BC** is perpendicular to **ADB** (meaning that **BC** is at a 90° angle to the line or plane containing the equilateral triangle).
- You are asked to find **∠ADC**.
- Given: **∠BCD = 54°**, and **AB** is part of the equilateral triangle.

To solve this, here's the reasoning:

### Step-by-Step Solution:
1. **Equilateral Triangle Properties**:
   Since **ADB** is an equilateral triangle, all angles in the triangle **ADB** are 60°. This means:
   - **∠ADB = ∠DAB = ∠ABD = 60°**.

2. **Perpendicular Line and Angle Calculation**:
   The problem mentions that **BC** is perpendicular to the plane or line containing the equilateral triangle. Since **∠BCD = 54°**, the total angle in triangle **ADC** (which includes **∠ADC**) will use the property that the sum of angles in a triangle is 180°.

3. **Sum of Angles in Triangle ADC**:
   The sum of angles in triangle **ADC** must be 180°. You have:
   - **∠ADC** (the angle we need to find),
   - **∠BCD = 54°**,
   - and some additional geometric reasoning based on the fact that triangle **ADB** is equilateral and perpendicularity conditions are involved.

Therefore, using geometric reasoning, you'd set up an equation that solves for **∠ADC** using the information given.

**Conclusion**: 
The exact calculation of **∠ADC** depends on correctly applying geometric relationships, particularly how the perpendicularity and equilateral triangle properties combine in this configuration. I can help you finalize the steps further if you need more clarity!

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Would you like to see further details on the geometric relationships, or do you have any other questions about this problem?

### 5 Related Questions:
1. What is the significance of the perpendicularity condition in this type of triangle problem?
2. How does the sum of angles in a triangle apply to solving for unknown angles?
3. What other properties of equilateral triangles are commonly used in geometry problems?
4. How can you approach solving problems involving both perpendicularity and equilateral triangles?
5. What is the relationship between angles when a triangle has one side perpendicular to another line?

### Tip:
Always remember to utilize the properties of specific triangles (like equilateral) fully, as they simplify many angle calculations!

Triangle ADB is an equilateral triangle. BCD = 54° and AB is perpendicular to BC. Find ∠ADC.

Learn how to find ∠ADC in an equilateral triangle with BCD = 54° and AB perpendicular to BC. This problem covers key geometry concepts, including equilateral triangle properties and the angle sum property of triangles. Suitable for students in Grades 8-10.

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