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Refer to the given diagram:

Note that, ﻿angle D B C equals 32 degree comma space angle A D B equals 22 degree comma space F i n d space angle A D C﻿





﻿54 degree﻿


﻿58 degree﻿


﻿36 degree﻿


﻿32 degree﻿

The diagram shows a circle with points \( A, B, C, D \) on its circumference and point \( O \) at the center. You are given the following angles:

- \( \angle DBC = 32^\circ \)
- \( \angle ADB = 22^\circ \)

We are asked to find \( \angle ADC \).

### Step-by-step Solution:

1. **Key Observations**:
   - \( ABCD \) is a cyclic quadrilateral (since all points lie on the circumference of a circle).
   - In any cyclic quadrilateral, the opposite angles sum up to \( 180^\circ \).

2. **Using Triangle Properties**:
   Consider the triangle \( \triangle ADB \). We know:
   - \( \angle ADB = 22^\circ \)
   - \( \angle DBC = 32^\circ \)
   - By the exterior angle theorem, in \( \triangle ABC \), the exterior angle \( \angle ADC \) is equal to the sum of the opposite interior angles \( \angle ADB + \angle DBC \).

3. **Calculation**:
   \[
   \angle ADC = \angle ADB + \angle DBC
   \]
   Substituting the given values:
   \[
   \angle ADC = 22^\circ + 32^\circ = 54^\circ
   \]

Thus, \( \angle ADC = 54^\circ \).

The correct answer is \( \boxed{54^\circ} \).

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Would you like more details or have any other questions?

### Related Questions:
1. What are the properties of cyclic quadrilaterals?
2. How does the exterior angle theorem apply to triangle problems in circles?
3. What is the relationship between an inscribed angle and the arc it subtends?
4. How can we find other angles in the diagram based on the information given?
5. What are the criteria for points to form a cyclic quadrilateral?

**Tip:** In any cyclic quadrilateral, opposite angles always sum to \( 180^\circ \). This is a useful property when solving problems involving circles and quadrilaterals!

Refer to the given diagram: Note that, ﻿angle D B C equals 32 degrees, angle A D B equals 22 degrees, Find angle A D C. Options: 54 degrees, 58 degrees, 36 degrees, 32 degrees.

Learn how to find angle ADC given angles DBC = 32 degrees and ADB = 22 degrees using properties of cyclic quadrilaterals and the exterior angle theorem. This solution includes step-by-step calculations and detailed explanations suitable for high school students in Grades 9-10.

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