Question

determine the value of x and identify which reason justifies your answer.
image of a triangle with angles 63°, 33°, and x°
the value of x is 30 because 33 + 30 equals 63 and 180 − 63 equals 117 and x is equal to the difference between 180 and the two remote interior angles.
the value of x is 30 because 180 − 33 equals 147 and x is equal to the difference between 180 and the larger of the two remote interior angles.
the value of x is 30 because 33 − 30 equals 3 and x is equal to the difference between 180 and the two remote interior angles.
the value of x is 30 because 33 + 30 equals 63 and the exterior angle shown is equal to the sum of the two remote interior angles.

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 (remote) interior angles. In the given triangle, the exterior angle (let's assume the angle related to the 63° and 33° angles) should be equal to the sum of the two remote interior angles. Wait, actually, looking at the triangle, we can also use the fact that in a triangle, the sum of an exterior angle and its adjacent interior angle is 180°, and also the sum of interior angles of a triangle is 180°. But more directly, for the angle \(x\), we can use the Exterior Angle Theorem or the fact that the sum of the two remote interior angles (33° and \(x\)) should equal the exterior angle (63°? Wait, no, let's re - examine. Wait, the triangle has angles 33°, \(x\), and the angle adjacent to 63°. Since 63° and its adjacent angle in the triangle are supplementary (they form a linear pair), the adjacent angle is \(180 - 63=117\)°. Then, using the sum of interior angles of a triangle (\(180^{\circ}\)), we have \(33 + x+117 = 180\). Simplifying, \(x + 150=180\), so \(x = 30\). But the correct reasoning is based on the Exterior Angle Theorem: the exterior angle (63°) is equal to the sum of the two remote interior angles (33° and \(x\))? Wait, no, 33 + 30 = 63. So the value of \(x\) is 30 because 33+30 equals 63 and the exterior angle shown is equal to the sum of the two remote interior angles.

Step2: Analyze the given options

Looking at the options, the last option says "The value of x is 30 because 33 + 30 equals 63 and the exterior angle shown is equal to the sum of the two remote interior angles." This is in line with the Exterior Angle Theorem which states that an exterior angle of a triangle is equal to the sum of the two non - adjacent (remote) interior angles.

Answer:

The value of \(x\) is 30 because 33 + 30 equals 63 and the exterior angle shown is equal to the sum of the two remote interior angles. (The corresponding option that states this reasoning)