Question

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Explanation:

Step1: Find the measure of the angle

Since the exterior angle is \(20^{\circ}\), the interior angle \(y = 180^{\circ}- 20^{\circ}=160^{\circ}\). But in a right - angled triangle, the sum of the two non - right angles is \(90^{\circ}\). Wait, no, actually, using the property of the exterior angle of a triangle. The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In a right - angled triangle (one angle is \(90^{\circ}\)), if the exterior angle is \(20^{\circ}\), we can also use trigonometric ratios. The side adjacent to the angle (the angle whose exterior is \(20^{\circ}\)) is \(10\), and we want to find the side \(x\) (opposite to the \(20^{\circ}\) angle). We know that \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). Here, \(\tan20^{\circ}=\frac{x}{10}\)

Step2: Solve for \(x\)

$$x = 10\times\tan20^{\circ}$$

Using a calculator, \(\tan20^{\circ}\approx0.364\)

$$x=10\times0.364 = 3.64$$

Answer:

\(x\approx3.64\)