Question

1. determine the value of x in the diagram.

40+30=70
180-70=110
x=110

Explanation:

Step1: Recall the exterior angle theorem

The exterior angle theorem states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In triangle \(ASM\), the exterior angle at \(M\) (which is \(\angle x\)) is equal to the sum of \(\angle A\) and \(\angle S\).

Step2: Identify the non - adjacent interior angles

We know that \(\angle A = 30^{\circ}\) and \(\angle S=42^{\circ}\).

Step3: Calculate the value of \(x\)

Using the exterior angle theorem, \(x=\angle A+\angle S\). Substituting the values of \(\angle A\) and \(\angle S\), we get \(x = 30^{\circ}+ 42^{\circ}=72^{\circ}\)? Wait, no, wait. Wait, maybe the previous wrong calculation. Wait, let's re - examine. Wait, the sum of interior angles of a triangle is \(180^{\circ}\). First, find the interior angle at \(M\) (let's call it \(\angle S MA\)). \(\angle S MA=180^{\circ}-\angle A-\angle S=180 - 30 - 42=108^{\circ}\)? No, no. Wait, the angle \(x\) and \(\angle S MA\) are supplementary (they form a linear pair). So \(\angle S MA = 180 - x\). And by the angle - sum property of a triangle, \(\angle A+\angle S+\angle S MA=180\). So \(30 + 42+(180 - x)=180\). Then \(72+180 - x=180\). Then \(252 - x=180\). Then \(x = 252 - 180 = 72\)? Wait, the original wrong calculation was wrong. Let's do it correctly.

Wait, the exterior angle theorem: the exterior angle is equal to the sum of the two remote interior angles. So the exterior angle at \(M\) (angle \(x\)) is equal to \(\angle A+\angle S\). So \(\angle A = 30^{\circ}\), \(\angle S = 42^{\circ}\), so \(x=30 + 42=72^{\circ}\). The previous calculation in the image was wrong. Let's correct it.

Step1: Apply the exterior angle theorem

The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles. In \(\triangle ASM\), \(\angle x\) is an exterior angle at \(M\), and the two non - adjacent interior angles are \(\angle A = 30^{\circ}\) and \(\angle S=42^{\circ}\).

Step2: Calculate \(x\)

By the exterior angle theorem, \(x=\angle A+\angle S\).
Substitute \(\angle A = 30^{\circ}\) and \(\angle S = 42^{\circ}\) into the formula: \(x=30^{\circ}+42^{\circ}=72^{\circ}\).

Wait, the original calculation in the image was wrong. Let's do it step by step correctly.

1. Recall the exterior angle theorem: For a triangle, an exterior angle is equal to the sum of the two interior angles that are not adjacent to it.
2. In \(\triangle ASM\), the exterior angle at \(M\) (angle \(x\)) is formed by extending side \(AM\) to \(K\). The two non - adjacent interior angles to \(\angle x\) are \(\angle A\) (at \(A\)) and \(\angle S\) (at \(S\)).
3. So, by the exterior angle theorem, \(x=\angle A+\angle S\).
4. Substitute \(\angle A = 30^{\circ}\) and \(\angle S = 42^{\circ}\): \(x = 30^{\circ}+42^{\circ}=72^{\circ}\).

Answer:

\(x = 72^{\circ}\)