To solve for the side $x$ of the right-angled triangle, we can use the sine function from trigonometry.

In a right-angled triangle, the sine of an angle is defined as:

$\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}$

Here:

• $\theta = 20^\circ$
• The hypotenuse is 80.
• The opposite side is $x$.

Thus, the equation becomes:

$\sin(20^\circ) = \frac{x}{80}$

To find $x$, multiply both sides by 80:

$x = 80 \times \sin(20^\circ)$

Using a calculator, we get:

$x \approx 80 \times 0.3420$ $x \approx 27.36$

So, the length of the side opposite the 20° angle is approximately 27.36 units.

Would you like further details or clarification on this? Here are some related questions you can explore:

1. How would the result change if the angle was 30° instead of 20°?
2. What if the hypotenuse had a different value, say 100?
3. How can we use the cosine or tangent functions for this kind of problem?
4. What is the relationship between the other angle (i.e., 70°) and the sides of the triangle?
5. How do we verify our result using the Pythagorean theorem?

Tip: When dealing with angles and sides in right-angled triangles, always check which trigonometric function (sine, cosine, tangent) is appropriate based on what you're given (opposite, adjacent, or hypotenuse).