Question

find the value of x.
(there is a triangle with an exterior angle of 145°, one interior angle of 48°, and the angle to find is x°)

Explanation:

Step1: Find the adjacent interior angle

The angle adjacent to \(145^\circ\) on a straight line is \(180^\circ - 145^\circ = 35^\circ\).

Step2: Use triangle angle sum

The sum of angles in a triangle is \(180^\circ\). So \(x = 180^\circ - 48^\circ - 35^\circ = 97^\circ\)? Wait, no, wait. Wait, the triangle: one angle is \(48^\circ\), the other non - right? Wait, no, wait, the angle adjacent to \(145^\circ\) is supplementary, so \(180 - 145 = 35^\circ\). Then in the triangle, angles sum to \(180\), so \(x=180 - 48 - 35\)? Wait, no, wait, maybe it's a right triangle? Wait, no, the diagram: the triangle has one angle \(48^\circ\), the exterior angle is \(145^\circ\). Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two non - adjacent interior angles. So \(145^\circ=48^\circ + x\)? Wait, no, that would be if it's the exterior angle. Wait, no, the angle adjacent to \(145^\circ\) is \(180 - 145 = 35^\circ\). Then in the triangle, the angles are \(48^\circ\), \(35^\circ\), and \(x\). Wait, but if it's a right triangle? No, the diagram shows a triangle with one angle \(48^\circ\), and the base angle adjacent to the \(145^\circ\) exterior angle. Wait, let's correct: the exterior angle of a triangle is equal to the sum of the two remote interior angles. So the \(145^\circ\) angle is an exterior angle, so it should equal \(48^\circ + x\)? Wait, no, that would be if \(x\) is one of the remote angles. Wait, no, the angle adjacent to \(145^\circ\) is \(180 - 145 = 35^\circ\), then in the triangle, the three angles are \(48^\circ\), \(35^\circ\), and \(x\). But if it's a right triangle? No, the diagram: let's assume it's a triangle where one angle is \(48^\circ\), the other angle (adjacent to the \(145^\circ\) exterior angle) is \(180 - 145 = 35^\circ\), then \(x = 180 - 48 - 35=97\)? Wait, no, that can't be. Wait, maybe the triangle is a right triangle? Wait, no, the diagram: the right angle? Wait, no, the user's diagram: maybe it's a triangle with an exterior angle of \(145^\circ\), one interior angle \(48^\circ\), and we need to find \(x\). Wait, the correct approach: the exterior angle is equal to the sum of the two non - adjacent interior angles. So \(145^\circ=48^\circ + x\)? No, that would be if \(x\) is the other non - adjacent angle. Wait, no, the angle adjacent to \(145^\circ\) is \(180 - 145 = 35^\circ\). Then in the triangle, \(48^\circ+35^\circ + x = 180^\circ\), so \(x = 180-(48 + 35)=97\)? Wait, but that seems odd. Wait, maybe I made a mistake. Wait, let's re - examine: the exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two remote (non - adjacent) interior angles. So if the exterior angle is \(145^\circ\), and one remote interior angle is \(48^\circ\), then the other remote interior angle \(x\) satisfies \(145 = 48 + x\)? No, that would mean \(x = 145 - 48 = 97\)? Wait, that's the same as before. Wait, maybe that's correct.

Wait, let's do it step by step:

Step 1: Find the adjacent interior angle

The angle adjacent to the \(145^\circ\) exterior angle is supplementary to it. So, let's call this angle \(y\). Then \(y+145^\circ = 180^\circ\), so \(y=180 - 145=35^\circ\).

Step 2: Use the triangle angle - sum property

In a triangle, the sum of the interior angles is \(180^\circ\). We know one angle is \(48^\circ\), another is \(y = 35^\circ\), and the third is \(x\). So \(48^\circ+35^\circ+x = 180^\circ\).

Step 3: Solve for \(x\)

First, add \(48\) and \(35\): \(48 + 35 = 83\). Then, \(x=180 - 83 = 97\)? Wait, no, \(48+35 = 83\), \(180 - 83 = 97\). Wait, but let's check with the exterior angle theorem. The exterior angle (\(145^\circ\)) should be equal to the sum of the two non - adjacent interior angles (\(48^\circ\) and \(x\)). So \(145=48 + x\), then \(x = 145 - 48=97\). Yes, that's correct.

Answer:

\(x = 97\)