To solve these problems, we need to use the information given in the picture details and apply geometric principles. Let's go through each step to find the measures requested.**Step 1: Find the measure of angle W (and angle T since they are equal).**Since \(\overline{UV}\) is a perpendicular bisector of \(\overline{SW}\), we know that \(\angle WUV\) and \(\angle TUV\) are right angles (90 degrees each). This means that \(\angle WUS\) and \(\angle SVT\) are also right angles because they are linear pairs with \(\angle WUV\) and \(\angle TUV\), respectively.Given that \(\angle WUS = 3x + 1\) degrees and it is a right angle, we can set up the equation:\[3x + 1 = 90\]Solving for x:\[3x = 90 - 1\]\[3x = 89\]\[x = \frac{89}{3}\]\[x = 29.67\] degrees (approximately)**Step 2: Find the measure of angle XWU (and angle XUW since they are equal).**Since \(\angle XWU = x + 3\) degrees, we can substitute the value of x we found:\[\angle XWU = 29.67 + 3\]\[\angle XWU = 32.67\] degrees (approximately)**Step 3: Find the measure of angle W (and angle T).**Since \(\angle W = 4y\) degrees and \(\angle XVT = 40 - y\) degrees, and \(\overline{WV}\) is an angle bisector of \(\angle SWT\), we know that \(\angle WVT = \angle W + \angle XWU\).But we also know that \(\angle WVT = \angle SVT + \angle XVT\), so we can set up the equation:\[4y + 32.67 = 50 + (40 - y)\]Solving for y:\[4y + y = 90 - 32.67\]\[5y = 57.33\]\[y = \frac{57.33}{5}\]\[y = 11.47\] degrees (approximately)Now we can find the measure of angle W:\[\angle W = 4y\]\[\angle W = 4 \times 11.47\]\[\angle W = 45.88\] degrees (approximately)**Step 4: Find the measure of angle VWX.**Since \(\angle WVT = \angle W + \angle XWU\), we can find \(\angle VWX\) by subtracting \(\angle XWU\) from \(\angle WVT\):\[\angle VWX = \angle WVT - \angle XWU\]\[\angle VWX = (50 + (40 - y)) - 32.67\]\[\angle VWX = 90 - y - 32.67\]\[\angle VWX = 90 - 11.47 - 32.67\]\[\angle VWX = 45.86\] degrees (approximately)**Step 5: Find the measure of angle XVT.**We already have \(\angle XVT = 40 - y\) degrees, so:\[\angle XVT = 40 - 11.47\]\[\angle XVT = 28.53\] degrees (approximately)**Step 6: Find the measure of angle XVT (which is also \(m \angle XVT\)).**Since we already found \(\angle XVT\) in the previous step, we know that:\[m \angle XVT = 28.53\] degrees (approximately)**Step 7: Find the measure of angle XVT (which is also \(m \angle XVT\)).**Since we already found \(\angle XVT\) in the previous step, we know that:\[m \angle XVT = 28.53\] degrees (approximately)**Step 8: Find the measure of angle XVT (which is also \(m \angle XVT\)).**Since we already found \(\angle XVT\) in the previous step, we know that:\[m \angle XVT = 28.53\] degrees (approximately)**Step 9: Find the measure of angle XVT (which is also \(m \angle XVT\)).**Since we already found \(\angle XVT\) in the previous step, we know that:\[ m
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