Okay, let's see. I need to find the measure of the largest angle in a convex quadrilateral where the largest angle is twice the smallest, and the other two angles are right angles. Hmm. Let me start by recalling some basics about quadrilaterals. 

First, the sum of the interior angles in any convex quadrilateral is 360 degrees. That's a key point. So, if I can figure out the measures of all four angles and relate them according to the problem, I should be able to solve for the unknowns.

The problem states that there are four angles in the quadrilateral. Two of them are right angles, which means they're each 90 degrees. Let's denote the smallest angle as 'x' degrees. Then, the largest angle is twice that, so it's '2x' degrees. The other two angles are right angles, so they're each 90 degrees. 

So, putting that together, the four angles are: x (smallest), 2x (largest), 90, and 90. Now, if I sum all these up, they should equal 360 degrees. Let me write that equation out:

x + 2x + 90 + 90 = 360

Simplifying the left side: x + 2x is 3x, and 90 + 90 is 180. So, the equation becomes:

3x + 180 = 360

Now, subtract 180 from both sides to solve for 3x:

3x = 360 - 180
3x = 180

Divide both sides by 3:

x = 60

So, the smallest angle is 60 degrees. Then, the largest angle, which is twice that, would be 2 * 60 = 120 degrees. 

Wait, let me double-check that. If the angles are 60, 120, 90, and 90, adding them up: 60 + 120 = 180, and 90 + 90 = 180. 180 + 180 = 360. Yep, that adds up. So, the largest angle is 120 degrees. 

But hold on, since it's a convex quadrilateral, all angles must be less than 180 degrees. Our largest angle is 120, which is indeed less than 180, so that's okay. Also, the smallest angle is 60, which is positive, so that's fine. The other two angles are 90, which are right angles. So, all conditions are satisfied. 

I think that's it. The answer should be 120 degrees. Let me just confirm once more. If x is 60, then 2x is 120. The two right angles are 90 each. 60 + 120 = 180, 90 + 90 = 180, 180 + 180 = 360. Perfect. 

I don't see any mistakes here. The logic seems solid. The key was remembering the sum of interior angles in a quadrilateral and setting up the equation correctly based on the given relationships between the angles. Then solving for x and computing the largest angle. 

So, the final answer is 120 degrees. That should be it.
Therefore, the measure of the largest angle in the convex quadrilateral is $\boxed{120}$ degrees.
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