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To solve for angle \(\theta\) in the isosceles triangle, we can use trigonometric relationships. Here's the step-by-step process:

### Given:
- The isosceles triangle has two equal sides of length \(19.3 \, \text{m}\) each.
- The base of the triangle is \(14.2 \, \text{m}\).
- \(\theta\) is the angle at the base, and the triangle includes a right angle formed by drawing a height from the top vertex to the midpoint of the base.

### Step 1: Break the base in half.
Since the height divides the base in half, each half of the base is:
\[
\frac{14.2}{2} = 7.1 \, \text{m}.
\]

### Step 2: Use the tangent function.
In the right triangle formed, the tangent of angle \(\theta\) is the ratio of the opposite side (half the base) to the adjacent side (the equal side of the triangle).

\[
\tan(\theta) = \frac{7.1}{19.3}.
\]

### Step 3: Calculate \(\theta\).
First, find the value of the tangent:
\[
\tan(\theta) = \frac{7.1}{19.3} \approx 0.36736.
\]

Now, take the inverse tangent (arctangent) to find \(\theta\):
\[
\theta = \tan^{-1}(0.36736) \approx 20.2^\circ.
\]

### Final Answer:
The size of angle \(\theta\) is approximately \(20^\circ\) to the nearest integer.

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Would you like further details or explanations on any part of this solution?

Here are 5 related questions for deeper exploration:
1. How do you use sine or cosine in solving for angles in triangles?
2. How would you solve if the triangle were not isosceles?
3. What if you only knew the perimeter? Could you still find \(\theta\)?
4. How can we use the Pythagorean theorem in isosceles triangles?
5. Can trigonometric functions help find the height of the triangle?

**Tip**: Always verify if a triangle is isosceles when solving for angles using trigonometry, as it simplifies calculations!

Work out the size of angle θ in the isosceles triangle below. Give your answer in degrees to the nearest integer.

This problem focuses on calculating the angle θ in an isosceles triangle using trigonometric functions. With given side lengths of 19.3 m and 14.2 m, the solution involves applying the tangent formula. Suitable for students in Grades 9-11.

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