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The problem shows a triangle \( \triangle ABC \) with a line \( ED \parallel BC \), and angles and expressions provided at various points.

### Step 1: Solve for \( x \) (18.1)
Since \( ED \parallel BC \), we can use the concept of corresponding angles in parallel lines.

1. Angle \( EAD = 3x - 15^\circ \)
2. Angle \( D_1 = 4x + 20^\circ \)
3. Angle \( A = 70^\circ \)

In a triangle, the sum of the angles equals \( 180^\circ \). Therefore, for \( \triangle AED \):
\[
\angle A + \angle EAD + \angle D_1 = 180^\circ
\]
Substitute the given angles:
\[
70^\circ + (3x - 15^\circ) + (4x + 20^\circ) = 180^\circ
\]
Simplify:
\[
70^\circ + 3x - 15^\circ + 4x + 20^\circ = 180^\circ
\]
\[
70^\circ - 15^\circ + 20^\circ + 7x = 180^\circ
\]
\[
75^\circ + 7x = 180^\circ
\]
Solve for \( x \):
\[
7x = 180^\circ - 75^\circ
\]
\[
7x = 105^\circ
\]
\[
x = \frac{105^\circ}{7}
\]
\[
x = 15^\circ
\]

### Step 2: Calculate the size of \( \hat{D}_1 \) (18.2)
Now that \( x = 15^\circ \), substitute into the expression for \( \hat{D}_1 = 4x + 20^\circ \):
\[
\hat{D}_1 = 4(15^\circ) + 20^\circ = 60^\circ + 20^\circ = 80^\circ
\]

### Final Answers:
- **18.1**: \( x = 15^\circ \)
- **18.2**: \( \hat{D}_1 = 80^\circ \)

Would you like a detailed explanation of the angle relationships? 

Here are 5 questions for further practice:
1. How can you prove the sum of angles in any triangle equals \( 180^\circ \)?
2. What are corresponding angles, and why are they equal when lines are parallel?
3. How do you solve for unknown angles in a quadrilateral?
4. What is the difference between alternate and corresponding angles?
5. Can the angles formed in a triangle ever exceed 180 degrees?

**Tip:** Always check the parallel lines and corresponding angles when dealing with transversals in triangles!

In ∆ABC below, ED || BC. Calculate: 18.1 the value of x; 18.2 the size of angle D₁.

This problem involves solving for x and angle D₁ in a triangle where a line is parallel to one side. Concepts include corresponding angles, parallel lines, and the sum of angles in a triangle. Suitable for Grades 9-10.

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