Question

a side of the triangle below has been extended to form an exterior angle of 160°. find the value of x.

Explanation:

Answer:

To find the value of \( x \), we use the property that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

First, we find the adjacent interior angle to the \( 160^{\circ} \) exterior angle. Since a straight line is \( 180^{\circ} \), the adjacent interior angle \( y \) (let's call it) satisfies \( y+ 160^{\circ}=180^{\circ} \), so \( y = 180^{\circ}- 160^{\circ}=20^{\circ} \).

Now, using the angle - sum property of a triangle (the sum of interior angles of a triangle is \( 180^{\circ} \)), we have \( x + 48^{\circ}+20^{\circ}=180^{\circ} \).

Or, using the exterior - angle property directly: the exterior angle (\( 160^{\circ} \)) is equal to the sum of the two non - adjacent interior angles (\( x \) and \( 48^{\circ} \))? Wait, no. Wait, the exterior angle is equal to the sum of the two remote interior angles. Wait, the adjacent angle to the exterior angle is \( 180 - 160=20^{\circ} \), then \( x+48^{\circ}+20^{\circ}=180^{\circ} \), so \( x=180-(48 + 20)=112^{\circ} \)? Wait, no, that's wrong. Wait, the exterior angle theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles.

So, if the exterior angle is \( 160^{\circ} \), and one of the non - adjacent interior angles is \( 48^{\circ} \), and the other is \( x \), then \( 160^{\circ}=x + 48^{\circ} \)? No, that's not right. Wait, the adjacent angle to the exterior angle is \( 180 - 160 = 20^{\circ} \). Then in the triangle, the sum of angles is \( 180^{\circ} \), so \( x+48^{\circ}+20^{\circ}=180^{\circ} \), so \( x=180-(48 + 20)=112^{\circ} \)? Wait, no, let's correct.

Wait, the exterior angle is formed by extending a side. So the two non - adjacent interior angles to the exterior angle are \( x \) and \( 48^{\circ} \), and the exterior angle is equal to their sum. Wait, no, the exterior angle and the adjacent interior angle are supplementary. So adjacent interior angle \( = 180 - 160=20^{\circ} \). Then the sum of the interior angles of the triangle is \( 180^{\circ} \), so \( x + 48^{\circ}+20^{\circ}=180^{\circ} \), so \( x=180-(48 + 20)=112^{\circ} \)? No, that's incorrect. Wait, no, the exterior angle theorem: the measure of an exterior angle of a triangle is equal to the sum of the measures of the two remote (non - adjacent) interior angles.

So if the exterior angle is \( 160^{\circ} \), and one remote interior angle is \( 48^{\circ} \), and the other is \( x \), then \( 160=x + 48 \)? No, that would give \( x = 112 \), but that can't be. Wait, no, I think I mixed up. Let's draw the triangle mentally. The exterior angle is at a vertex, and the two non - adjacent angles are the other two angles of the triangle. Wait, the adjacent angle to the exterior angle is \( 180 - 160 = 20^{\circ} \). Then the triangle has angles \( x \), \( 48^{\circ} \), and \( 20^{\circ} \). So \( x+48 + 20=180 \), so \( x=180 - 68 = 112 \)? No, that's wrong. Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So the exterior angle (\( 160^{\circ} \)) should be equal to \( x+48^{\circ} \)? But \( 160-48 = 112 \), but that would mean the angle \( x \) is \( 112^{\circ} \), and the adjacent angle is \( 20^{\circ} \), and \( 112 + 48+20=180 \), which is correct. Wait, yes, that's right.

So \( x=160 - 48=112 \)? Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \( 160^{\circ} \), and one non - adjacent angle is \( 48^{\circ} \), and the other is \( x \), then \( 160=x + 48 \), so \( x = 160 - 48=112 \). And then the adjacent angle is \( 180 - 160 = 20^{\circ} \), and \( 112+48 + 20=180 \), which checks out.

So the value of \( x \) is \( 112 \).