Okay, so I have this geometry problem here. Let me try to figure it out step by step. The problem says: "A triangle triangle ABC with angle A = 14 degrees is inscribed in a circle where AB is a diameter. How many degrees are in angle B?" Hmm, let me visualize this first.

First, since triangle ABC is inscribed in a circle and AB is the diameter, that must mean that AB is the hypotenuse of the triangle. Wait, no, if AB is the diameter, then the circle is the circumcircle of the triangle. In a circle, if a triangle is inscribed and one of its sides is the diameter, then the triangle is a right-angled triangle. Is that a theorem? Yes, Thales' theorem states that if you have a triangle inscribed in a circle where one side is the diameter, then the angle opposite that side is a right angle. So, that means angle C, which is opposite AB, should be 90 degrees. Let me confirm that. If AB is the diameter, then angle C is the right angle. So, angle C is 90 degrees. That makes sense.

Given that, the triangle ABC has angles A = 14 degrees, angle C = 90 degrees, and angle B is what we need to find. Since the sum of the angles in a triangle is always 180 degrees, we can add up the known angles and subtract from 180 to find angle B. Let me write that down:

Angle A + Angle B + Angle C = 180 degrees
14 + Angle B + 90 = 180

So, adding 14 and 90 gives 104. Then, 180 - 104 = 76. Therefore, angle B should be 76 degrees. Wait, that seems straightforward. But let me make sure I didn't skip any steps or make any wrong assumptions.

Wait, is angle C necessarily the right angle? Because AB is the diameter, so the angle opposite AB is angle C. So, yes, angle C is 90 degrees. So, the triangle is right-angled at C. Therefore, angles A and B are the other two angles. Since angle A is 14 degrees, angle B must be 180 - 90 - 14 = 76 degrees. That checks out.

But let me think again. Maybe I should draw the triangle to visualize. Let's say we have a circle with diameter AB. Point C is somewhere on the circumference. Then triangle ABC is formed with AB as the diameter. By Thales' theorem, angle ACB is 90 degrees. So, angle C is 90 degrees. Then, angles at A and B must add up to 90 degrees. Since angle A is 14 degrees, angle B must be 76 degrees. That seems correct.

Is there another way to approach this problem? Maybe using the properties of circles and inscribed angles. Let me recall that the measure of an inscribed angle is half the measure of its intercepted arc. So, angle A is 14 degrees, which is an inscribed angle intercepting arc BC. Therefore, the measure of arc BC is 2 * 14 = 28 degrees. Similarly, angle C is 90 degrees, which intercepts arc AB. The measure of arc AB is 2 * 90 = 180 degrees. Wait, arc AB is the diameter, so it's 180 degrees, which makes sense. Then, the remaining arc AC would be the rest of the circle. The total circumference is 360 degrees, so arc AC = 360 - arc AB - arc BC = 360 - 180 - 28 = 152 degrees. Then, angle B is an inscribed angle intercepting arc AC. Therefore, angle B = 1/2 * arc AC = 1/2 * 152 = 76 degrees. That's the same answer as before. So, this method also confirms that angle B is 76 degrees.

Therefore, both methods give the same result. That makes me more confident that the answer is indeed 76 degrees. Let me check if there's any other angle or property I might have missed. For example, could there be a case where the triangle is not right-angled? But according to Thales' theorem, if AB is the diameter, then angle C must be 90 degrees. So, that's fixed. Then, the other angles are determined by the given angle A. So, angle B is uniquely determined as 76 degrees.

Alternatively, if I use the sine law, maybe I can verify. In triangle ABC, using the sine formula: a / sin A = b / sin B = c / sin C. But since AB is the diameter, the length of AB is 2R, where R is the radius of the circle. But since we don't have the actual lengths, maybe this approach is more complicated. However, let's try.

Let AB = 2R. Then, in triangle ABC, AB is the hypotenuse of the right-angled triangle, so AB = 2R. Then, side AC is opposite angle B, which is 76 degrees, and side BC is opposite angle A, which is 14 degrees. Using sine law: AC / sin B = AB / sin C. Since angle C is 90 degrees, sin C = 1. So, AC = AB * sin B = 2R * sin 76. Similarly, BC = AB * sin A = 2R * sin 14. Then, using Pythagoras theorem: AC² + BC² = AB². Let's check:

(2R sin 76)² + (2R sin 14)² = (2R)²
4R² sin² 76 + 4R² sin² 14 = 4R²
Divide both sides by 4R²:
sin² 76 + sin² 14 = 1

Is this true? Let me compute sin² 76 + sin² 14. Let's see:

sin 76 ≈ 0.9703, so sin² 76 ≈ 0.9414
sin 14 ≈ 0.2419, so sin² 14 ≈ 0.0585
Adding them: 0.9414 + 0.0585 ≈ 0.9999, which is approximately 1. So, due to rounding errors, it's close enough. Therefore, this also confirms that the calculations are consistent.

Therefore, all methods point to angle B being 76 degrees. I think that's solid. I don't see any mistakes in the reasoning. So, the answer should be 76 degrees.

**Final Answer**
The measure of angle B is \boxed{76} degrees.
Therefore, the measure of $\angle B$ is $\boxed{76}$ degrees.
</think>