Question

find x.
45°
59°
x°
x =

Explanation:

Step1: Identify triangle angle sum

A triangle's interior angles sum to \(180^\circ\). First, find the third angle of the triangle. The given angles are \(45^\circ\) (vertical angle, so equal to the angle inside the triangle) and \(59^\circ\). Let the third angle be \(y\). Then \(45^\circ + 59^\circ + y = 180^\circ\).

Step2: Calculate the third angle

Solve for \(y\): \(y = 180^\circ - 45^\circ - 59^\circ = 76^\circ\).

Step3: Use linear pair for \(x\)

The angle \(y\) and \(x\) form a linear pair (supplementary angles), so \(x + y = 180^\circ\). Substitute \(y = 76^\circ\): \(x = 180^\circ - 76^\circ = 104^\circ\)? Wait, no—wait, maybe I mixed up. Wait, the \(45^\circ\) is a vertical angle, so the angle inside the triangle is \(45^\circ\), the other angle is \(59^\circ\), so the third angle (let's call it \(z\)) is \(180 - 45 - 59 = 76\). But then \(x\) and \(z\) are same - side? Wait, no, actually, the angle adjacent to \(x\) is equal to the third angle of the triangle (corresponding angles or vertical angles). Wait, maybe a better approach: the exterior angle theorem. The exterior angle \(x\) is equal to the sum of the two non - adjacent interior angles. The two non - adjacent interior angles are \(45^\circ\) and \(59^\circ\). So \(x = 45^\circ+ 59^\circ\).

Step4: Calculate \(x\)

\(x = 45 + 59 = 104\)? Wait, no, wait. Wait, the exterior angle theorem: 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 we consider the triangle, the angle \(x\) is an exterior angle, and the two non - adjacent interior angles are \(45^\circ\) and \(59^\circ\). So \(x=45 + 59 = 104\)? Wait, no, wait, maybe I got the diagram wrong. Wait, looking at the diagram, the two angles inside the triangle (non - adjacent to \(x\)) are \(45^\circ\) (vertical angle) and \(59^\circ\). So by exterior angle theorem, \(x = 45+59 = 104\)? Wait, no, wait, 45 + 59 is 104? 45+59 = 104? Yes, 45+59 = 104. Wait, but let's check again. The triangle has angles: one is \(45^\circ\) (since vertical angles are equal), one is \(59^\circ\), so the exterior angle \(x\) is equal to the sum of these two, so \(x = 45 + 59=104\). Wait, no, wait, maybe I made a mistake. Wait, 45 + 59 is 104? 45+50 = 95, 95 + 9 = 104. Yes. So \(x = 45+59 = 104\).

Answer:

\(x = 104\)