https://file.aiquickdraw.com/mathbot/file/17285974281813fwfesnp.jpg

To solve for the size of angle \( \angle ADE \), let's analyze the given information in the diagram:

1. **Label Information:**
   - Angle \( \angle ABC = x + 10^\circ \).
   - Angle \( \angle BCD = 2x - 20^\circ \).

2. Since the diagram seems to form a cyclic quadrilateral (a quadrilateral where all vertices lie on a circle), we can use the property that the opposite angles of a cyclic quadrilateral sum to \( 180^\circ \).

3. If we assume that quadrilateral \( ABCD \) is cyclic, we can apply the following relation:
   \[
   \angle ABC + \angle ADE = 180^\circ
   \]
   \[
   \angle BCD + \angle DAB = 180^\circ
   \]

Now let's proceed with the calculation steps:

### Step 1: Set up the Equation for Angle Sum
We have:
\[
\angle ABC + \angle ADE = 180^\circ
\]
Substituting the values:
\[
(x + 10^\circ) + \angle ADE = 180^\circ
\]
\[
\angle ADE = 180^\circ - (x + 10^\circ)
\]
\[
\angle ADE = 170^\circ - x
\]

### Step 2: Analyze the Relationship of \( x \) Using the Other Given Angle
We also know the opposite angle relation:
\[
\angle BCD + \angle DAB = 180^\circ
\]
Let \( \angle DAB = y \). Then:
\[
(2x - 20^\circ) + y = 180^\circ
\]
\[
y = 200^\circ - 2x
\]

However, we need more specific information to find the exact value of \( x \) unless further conditions or information are provided in the problem.

### Conclusion
Without a specific value for \( x \), the expression for \( \angle ADE \) simplifies to:
\[
\angle ADE = 170^\circ - x
\]

Do you want details on how to derive \( x \) or further clarifications?

Here are 5 questions to deepen your understanding:
1. What properties distinguish a cyclic quadrilateral from a general quadrilateral?
2. How can angle relationships in cyclic quadrilaterals help in solving for unknown angles?
3. What are some examples of other geometric figures where opposite angles have similar properties?
4. Can the angle sum properties of triangles be used to further analyze quadrilaterals?
5. How can you determine if a quadrilateral is cyclic based on its angles?

**Tip:** Remember, in any cyclic quadrilateral, the sum of opposite angles always equals \( 180^\circ \). This property is a powerful tool for solving angle-related problems.

Work out the size of angle ADE. Give reasons for your working.

Learn how to solve for angle ADE using the properties of cyclic quadrilaterals. This problem covers key geometric concepts like the sum of opposite angles in cyclic quadrilaterals and the Cyclic Quadrilateral Theorem. Suitable for students in Grades 9-10.

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