Question

. find the value of $n$.

1. $n^{circ}=?$

Explanation:

Step1: Find the adjacent angle of \(130^{\circ}\)

The sum of adjacent angles on a straight line is \(180^{\circ}\). Let the adjacent angle be \(x\), then \(x = 180^{\circ}-130^{\circ}=50^{\circ}\)

Step2: Find the adjacent angle of \(64^{\circ}\)

Let the adjacent angle be \(y\), then \(y = 180^{\circ}-64^{\circ}=116^{\circ}\)

Step3: Use the triangle angle - sum property

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(n\). We know two angles of the triangle: \(x = 50^{\circ}\) and \(y = 116^{\circ}\) (This is wrong, actually, we should use the exterior - angle property. Let's correct it.)

Correct Step1: Use the exterior - angle property of a triangle
The exterior - angle property of a triangle states that an exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
The exterior angle adjacent to \(n\) is \(130^{\circ}\), and one non - adjacent interior angle is \(180 - 64=116^{\circ}\) (wrong again. Let's start over.)

Correct Step1: Use the exterior - angle property
The exterior angle of a triangle is equal to the sum of the two non - adjacent interior angles.
The exterior angle of \(130^{\circ}\) has two non - adjacent interior angles: one is \(n\) and the other is \(180 - 64 = 116^{\circ}\) (no, wrong. Let's use the correct formula)

Let's use the formula: If we consider the linear pair, the interior angle adjacent to \(130^{\circ}\) is \(a=180 - 130=50^{\circ}\), and the interior angle adjacent to the other exterior ray (related to \(64^{\circ}\)) is \(b = 180-64 = 116^{\circ}\) (no, wrong. Let's use the triangle angle sum. Let the three interior angles of the triangle be \(A\), \(B\), \(n\). If one exterior angle is \(130^{\circ}\), then \(A = 180 - 130=50^{\circ}\), another exterior - related angle: assume the other exterior angle (not in the figure's marked \(n\) part) has an adjacent interior angle \(B\). But we can also use the formula \(n+50=116\) (where \(116 = 180 - 64\)) (no).

Correct approach:
The sum of angles on a straight line: the interior angle adjacent to \(130^{\circ}\) is \(180 - 130=50^{\circ}\), the interior angle adjacent to the other (not \(n\)) is \(180 - 64 = 116^{\circ}\) (no). Wait, no. Let's use the triangle angle sum. The sum of angles in a triangle is \(180^{\circ}\).
Let the three angles of the triangle be \(x\), \(y\), \(n\). We know that \(x = 180 - 130=50^{\circ}\), \(y=180 - 64 = 116^{\circ}\) (no, wrong. Wait, no. The two angles adjacent to the exterior angles and \(n\) form a triangle.
The two angles inside the triangle (not \(n\)): one is \(180 - 130 = 50^{\circ}\), another is \(180-(180 - 64)=64^{\circ}\) (using the property that the angle adjacent to an exterior angle and the exterior angle are supplementary).
By the triangle angle sum formula \(n+50 + 64=180\)

Step1: Calculate the sum of two known interior angles

The two interior angles of the triangle (found from the supplementary angles of the given exterior angles) are \(a = 180-130 = 50^{\circ}\) and \(b=64^{\circ}\) (because the angle adjacent to \(64^{\circ}\) exterior angle and \(64^{\circ}\) are supplementary, but in the triangle, the angle is \(64^{\circ}\))

Step2: Use the triangle angle - sum formula

The sum of angles in a triangle is \(180^{\circ}\). Let \(n\) be the unknown angle. Then \(n+50 + 64=180\)

$$n=180-(50 + 64)$$

$$n=180 - 114$$

$$n = 66^{\circ}$$

Answer:

\(66^{\circ}\)