Question

find x.
(image of a geometric figure with angles 97°, 30°, 35° and angle x° to find)
x =

Explanation:

Step1: Identify exterior angle property

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. But first, let's find the relevant angles. The angle of \(97^{\circ}\) and the angle adjacent to \(x\) and \(30^{\circ}\) are related. Wait, actually, we can use the fact that the sum of angles in a triangle and the linear pair. Wait, another approach: The exterior angle (the \(97^{\circ}\) angle) should be equal to the sum of the two remote interior angles of the smaller triangle? Wait, no. Let's look at the angles. The angle at the bottom right, the \(35^{\circ}\) angle and the angle inside the triangle: the angle inside the triangle (adjacent to \(35^{\circ}\)) is equal to \(35^{\circ}\) (vertical angles? No, alternate exterior? Wait, no, the \(35^{\circ}\) is an exterior angle, so the interior angle at that vertex is \(180 - 35=145^{\circ}\)? No, that's not right. Wait, maybe using the exterior angle theorem. Wait, the \(97^{\circ}\) angle is an exterior angle of the triangle that has angles \(x\), \(30^{\circ}\), and the angle equal to \(35^{\circ}\)? Wait, no. Wait, let's think again. The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. So, for the triangle with angle \(x\), \(30^{\circ}\), and the angle that is equal to \(97^{\circ}- 35^{\circ}\)? No, wait, the \(97^{\circ}\) is an exterior angle, and the two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? Wait, no. Wait, the \(35^{\circ}\) angle and the angle inside the triangle (let's call it \(y\)) are vertical angles? No, the \(35^{\circ}\) is an exterior angle, so the interior angle at that vertex is equal to \(35^{\circ}\) (if they are alternate angles? Wait, maybe the correct approach is: The exterior angle \(97^{\circ}\) is equal to \(x + 30^{\circ}+35^{\circ}\)? No, that can't be. Wait, no, let's calculate the angle at the third vertex of the triangle containing \(x\) and \(30^{\circ}\). The angle at the third vertex is equal to \(35^{\circ}\) (because of the alternate exterior angles or vertical angles? Wait, the \(35^{\circ}\) angle and the angle inside the triangle (at the intersection of the two lines) are equal. So, in the triangle with angles \(x\), \(30^{\circ}\), and \(35^{\circ}\), the exterior angle is \(97^{\circ}\). So by the exterior angle theorem, \(x + 30^{\circ}+35^{\circ}=97^{\circ}\)? Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(97^{\circ}\), and the two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? No, that's not correct. Wait, let's do it step by step.

First, find the angle inside the triangle that is opposite to the \(35^{\circ}\) angle. Since the \(35^{\circ}\) angle and that interior angle are equal (vertical angles? No, alternate angles). Wait, the two horizontal lines are parallel? No, they are the same line (a straight line). Wait, the \(35^{\circ}\) angle is an exterior angle, so the interior angle at that vertex is \(35^{\circ}\) (because it's a corresponding angle? Wait, maybe the triangle has angles \(x\), \(30^{\circ}\), and \(35^{\circ}\), and the exterior angle is \(97^{\circ}\). So by the exterior angle theorem, \(x+30^{\circ}+35^{\circ}=97^{\circ}\)? No, the exterior angle is equal to the sum of the two non - adjacent interior angles. So if the exterior angle is \(97^{\circ}\), and the two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? No, that's adding three angles. Wait, no, the exterior angle is equal to the sum of the two non - adjacent interior angles. So the two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? No, that's a mistake. Wait, the correct formula is: Exterior angle=\(x + (30^{\circ}+35^{\circ})\)? No, wait, let's calculate \(x\) as follows:

We know that in a triangle, the sum of interior angles is \(180^{\circ}\), and the exterior angle is \(180^{\circ}-\) interior angle. Wait, the exterior angle \(97^{\circ}\) is equal to \(x + 30^{\circ}+35^{\circ}\)? No, let's solve for \(x\) using the equation:

\(x+30 + 35=97\)

Step2: Solve for \(x\)

\(x+65 = 97\)

Subtract 65 from both sides:

\(x=97 - 65\)

\(x = 32\)? Wait, no, that's not right. Wait, maybe the angle at the third vertex is not \(35^{\circ}\). Wait, maybe the \(97^{\circ}\) is an exterior angle, and the two non - adjacent interior angles are \(x\) and \(30^{\circ}\), and the third angle is \(35^{\circ}\). Wait, no, let's re - examine the diagram. The \(35^{\circ}\) angle is an exterior angle, so the interior angle at that vertex is \(180 - 35=145^{\circ}\)? No, that's incorrect. Wait, I think I made a mistake. Let's use the exterior angle theorem correctly. The exterior angle ( \(97^{\circ}\)) is equal to the sum of the two non - adjacent interior angles of the triangle. The two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? No, the \(35^{\circ}\) is an exterior angle, so the interior angle is \(35^{\circ}\) (if they are alternate angles). Wait, maybe the correct equation is \(x + 30=97 - 35\)

Let's calculate \(97-35 = 62\)

Then \(x+30 = 62\)

Subtract 30 from both sides: \(x=62 - 30=32\)? No, that's not right. Wait, maybe the angle at the bottom is \(35^{\circ}\), and the angle inside the triangle is \(35^{\circ}\), so the sum of the interior angles of the triangle is \(x + 30+35\), and the exterior angle is \(97^{\circ}\), so \(x + 30+35=97\), so \(x=97-(30 + 35)=97 - 65 = 32\). Wait, but that seems low. Wait, maybe I messed up the angles. Wait, the \(97^{\circ}\) is an exterior angle, so the interior angle adjacent to it is \(180 - 97 = 83^{\circ}\). Then, in the triangle, the sum of angles is \(180^{\circ}\), so \(83+x + 30+35=180\)? No, that's not right. Wait, no, the triangle has three angles: \(x\), \(30^{\circ}\), and the angle equal to \(35^{\circ}\) (because of the vertical or alternate angles). So \(x+30 + 35=180 - (180 - 97)\)? No, this is getting confusing. Wait, let's start over.

The exterior angle theorem: An exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.

Looking at the diagram, the exterior angle is \(97^{\circ}\), and the two non - adjacent interior angles are \(x\) and \(30^{\circ}+35^{\circ}\)? No, the \(35^{\circ}\) is an exterior angle, so the interior angle is \(35^{\circ}\). So the two non - adjacent interior angles to the \(97^{\circ}\) exterior angle are \(x\) and \(30^{\circ}+35^{\circ}\)? No, the \(97^{\circ}\) is outside the triangle, and the two non - adjacent interior angles are \(x\) and the angle that is \(30^{\circ}+35^{\circ}\). Wait, no, the angle inside the triangle at the bottom is \(35^{\circ}\) (since the \(35^{\circ}\) is an exterior angle, and they are equal, maybe alternate exterior angles). So the triangle has angles \(x\), \(30^{\circ}\), and \(35^{\circ}\), and the exterior angle is \(97^{\circ}\). So by exterior angle theorem, \(x + 30+35=97\), so \(x=97 - 65 = 32\). Wait, but I think I was wrong earlier. Wait, let's check with the sum of angles in a triangle. If \(x = 32\), \(30\), and \(35\), sum is \(32 + 30+35=97\), and the exterior angle is equal to the sum of the two non - adjacent interior angles? No, the exterior angle is equal to the sum of the two non - adjacent interior angles. Wait, no, the exterior angle is supplementary to the adjacent interior angle. So the adjacent interior angle is \(180 - 97 = 83\). Then, in the triangle, \(83+x + 30+35=180\)? No, that's four angles. I think I made a mistake in identifying the triangle. The correct triangle is the one with angle \(x\), \(30^{\circ}\), and the angle that is equal to \(97^{\circ}-35^{\circ}\)? No, \(97 - 35 = 62\). Then \(x+30=62\), so \(x = 32\). Yes, that makes sense. So \(x=32\).

Answer:

\(x = 32\)