apply: circle problem #8\nab is a diameter of this circle.\nwhat is the value of $\\angle x^\\circ$?

Explanation:

Step1: Recall circle theorems

We know that an angle inscribed in a semicircle is a right angle, but here we use the property that the angle subtended by an arc at the center is twice the angle subtended at the circumference. Also, we can use the fact that in triangle or using the property of angles in the same segment. Wait, another approach: the angle between a tangent and a chord is equal to the angle in the alternate segment, but here we have a diameter. Wait, the given angle is (48^\circ), and we need to find (x). Let's see, the angle at the center and the angle at the circumference. Wait, actually, the triangle (ABC) or maybe the angle (x) and the (48^\circ) angle: since (AB) is a diameter, (\angle ACB = 90^\circ) (angle in a semicircle). But also, the angle between the chord and the diameter: wait, the key is that the angle (x) and the (48^\circ) angle: since the angle subtended by an arc at the center is (2x) (if (x) is at the circumference), but wait, the given angle is (48^\circ), maybe it's a vertical angle or supplementary? Wait, no, let's think again. The angle between the diameter (AB) and the chord (AD) is (x), and the angle at the center for the same arc is (48^\circ)? Wait, no, the angle marked (48^\circ) is at the center? Wait, the diagram shows a circle with center (O), (AB) is diameter, (A), (B), (C), (D) on the circle. The angle between two chords at the center? Wait, no, the angle marked (48^\circ) is between two chords, maybe at the center? Wait, actually, the correct approach is: the angle (x) is an inscribed angle, and the angle of (48^\circ) is related. Wait, the angle between the diameter (AB) and chord (AC) is (x), and the angle between chord (AD) and the other chord is (48^\circ). Wait, another way: in a circle, the angle between a chord and a diameter is equal to the angle in the alternate segment. Wait, no, let's use the fact that the sum of angles in a triangle or the property that the angle (x) and (48^\circ) are complementary? Wait, no, the correct answer is that (x = 90^\circ - 48^\circ = 42^\circ)? Wait, no, wait. Wait, the angle at the center is (48^\circ), so the angle at the circumference would be half? No, wait, the angle (x) is equal to (90^\circ - 48^\circ)? Wait, no, let's recall that in a circle, if we have a diameter (AB), and a chord (AC), then (\angle CAB = x), and the angle opposite or related. Wait, actually, the key is that the angle between the diameter and the chord is equal to the angle in the alternate segment, but here we have a right angle? Wait, no, let's do it step by step.

Wait, the angle marked (48^\circ) is the angle between two chords, maybe at the center, so the arc it subtends is (48^\circ), then the inscribed angle subtended by the same arc would be (24^\circ)? No, wait, no: the angle at the center is twice the angle at the circumference. Wait, maybe the angle (x) and the (48^\circ) angle: since (AB) is a diameter, (\angle ACB = 90^\circ) (angle in semicircle). Then, if we consider triangle (ABC), but maybe the other angle. Wait, no, the correct answer is that (x = 90^\circ - 48^\circ = 42^\circ)? Wait, no, let's think again. The angle between the diameter (AB) and chord (AD) is (x), and the angle between chord (AD) and another chord is (48^\circ), and since (AB) is diameter, the angle (\angle ADB = 90^\circ) (angle in semicircle). Wait, no, maybe the angle (x) is equal to (90^\circ - 48^\circ = 42^\circ)? Wait, no, let's check the circle theorem: the angle between a chord and a diameter is equal to the angle in the alternate segment. Wait, the angle (x) and the angle of (48^\circ): since the angle at the center is (48^\circ), the inscribed angle would be (24^\circ)? No, I think I made a mistake. Wait, the correct approach is: the angle marked (48^\circ) is an angle at the center, so the arc it subtends is (48^\circ), then the angle at the circumference subtended by the same arc is (24^\circ)? No, that's not right. Wait, no, the angle at the center is twice the angle at the circumference. So if the angle at the center is (48^\circ), the angle at the circumference is (24^\circ)? But that doesn't seem. Wait, maybe the angle (x) is equal to (90^\circ - 48^\circ = 42^\circ). Wait, let's think of the triangle: (AB) is diameter, so (\angle ACB = 90^\circ). The angle between (AC) and (AB) is (x), and the angle between (AC) and another chord is (48^\circ), so (x + 48^\circ = 90^\circ), so (x = 90 - 48 = 42). Yes, that makes sense. So (x = 42^\circ).

Step2: Calculate (x)

Since (AB) is a diameter, (\angle ACB = 90^\circ) (angle in a semicircle). The angle marked (48^\circ) and (x) are complementary (they add up to (90^\circ)) because they are angles in the right triangle (ACB) (or related to the right angle). So:

(x + 48^\circ = 90^\circ)

Solving for (x):

(x = 90^\circ - 48^\circ = 42^\circ)

Answer:

(42)