Question

find the measure indicated.

1. find ( mangle s )

Explanation:

Step1: Identify properties of a rhombus

In a rhombus, opposite angles are equal, and adjacent angles are supplementary (sum to \(180^\circ\)). Also, opposite angles are equal, so \(\angle V=\angle T\) and \(\angle S=\angle U\)? Wait, no, wait. Wait, in a rhombus, opposite angles are equal, and adjacent angles are supplementary. Wait, looking at the diagram, sides are marked as equal, so it's a rhombus. So \(\angle V\) and \(\angle T\) should be equal? Wait, the angles at \(V\) is \(8x\), at \(T\) is \(7x + 11\). So set them equal: \(8x=7x + 11\).

Step2: Solve for \(x\)

\(8x=7x + 11\)
Subtract \(7x\) from both sides: \(x = 11\).

Step3: Find measure of \(\angle V\) and \(\angle T\)

Now, \(\angle V=8x=8\times11 = 88^\circ\), \(\angle T=7x + 11=7\times11+11 = 88^\circ\) (checks out). Now, in a rhombus, adjacent angles are supplementary. So \(\angle S\) and \(\angle U\) are adjacent to \(\angle V\) and \(\angle T\)? Wait, \(\angle U\) is \(98^\circ\). Wait, no, in a rhombus, opposite angles are equal. Wait, maybe I mixed up. Wait, \(\angle U\) is \(98^\circ\), so its opposite angle \(\angle S\)? Wait, no, wait the diagram: vertices are \(S\), \(T\), \(U\), \(V\) in order? So \(S\) connected to \(T\) and \(V\), \(T\) connected to \(S\) and \(U\), \(U\) connected to \(T\) and \(V\), \(V\) connected to \(U\) and \(S\). So the angles: \(\angle S\) is opposite \(\angle U\)? Wait, no, in a quadrilateral, opposite angles are \(\angle S\) and \(\angle U\), \(\angle V\) and \(\angle T\). Wait, but \(\angle U\) is \(98^\circ\), so if \(\angle S\) is opposite \(\angle U\), then they should be equal? But that contradicts. Wait, no, maybe I got the order wrong. Wait, adjacent angles: \(\angle S\) is adjacent to \(\angle V\) and \(\angle T\), and \(\angle U\) is adjacent to \(\angle V\) and \(\angle T\). Wait, in a rhombus, adjacent angles are supplementary. So \(\angle S + \angle V=180^\circ\), \(\angle S + \angle T=180^\circ\), \(\angle U + \angle V=180^\circ\), \(\angle U + \angle T=180^\circ\). Wait, \(\angle U\) is \(98^\circ\), so \(\angle S\) should be supplementary to \(\angle U\)? Wait, no, opposite angles in a rhombus are equal. Wait, maybe the diagram has \(\angle U = 98^\circ\), so its opposite angle is \(\angle S\)? But that would mean \(\angle S = 98^\circ\), but that doesn't fit with adjacent angles. Wait, no, I think I made a mistake. Wait, let's re-examine. The quadrilateral is a rhombus (all sides equal). In a rhombus, opposite angles are equal, and adjacent angles are supplementary. So if \(\angle U = 98^\circ\), then its opposite angle (say \(\angle S\)) should be equal? No, wait no: in a rhombus, opposite angles are equal, adjacent angles are supplementary. Wait, maybe the angles at \(V\) and \(T\) are equal (since they are opposite), and angles at \(S\) and \(U\) are equal (opposite). Then adjacent angles: \(\angle V + \angle S=180^\circ\), \(\angle S + \angle T=180^\circ\), etc. Wait, we found \(\angle V = 88^\circ\), so \(\angle S\) (adjacent to \(\angle V\)) should be \(180^\circ - 88^\circ = 92^\circ\)? Wait, but \(\angle U\) is \(98^\circ\), which would be opposite \(\angle S\), but \(92 eq98\). Wait, that's a problem. Wait, maybe I misidentified the opposite angles. Wait, let's look at the diagram again. The sides: \(SV\) and \(ST\) are marked with two ticks? Wait, no, the diagram: \(SV\) and \(ST\) have two ticks? Wait, no, the original diagram: \(SV\) and \(ST\) are marked with two ticks? Wait, maybe the sides \(SV\) and \(VT\)? No, the user's diagram: vertices \(S\), \(T\), \(U\), \(V\). \(SV\) and \(ST\) have two ticks? Wait, no, the sides: \(SV\) and \(TU\) have one tick, \(ST\) and \(VU\) have two ticks? Wait, maybe it's a rhombus, so all sides equal. So angles at \(V\) and \(T\) are equal (opposite), angles at \(S\) and \(U\) are equal (opposite). Then \(\angle V = \angle T\), \(\angle S = \angle U\). Wait, but \(\angle U\) is \(98^\circ\), so \(\angle S = 98^\circ\), and \(\angle V = \angle T = 180 - 98 = 82^\circ\), but that contradicts our earlier calculation. Wait, no, we set \(\angle V = \angle T\) as \(8x\) and \(7x + 11\), solved \(x = 11\), got \(\angle V = 88\), \(\angle T = 88\). Then \(\angle S = \angle U\), and \(\angle V + \angle S = 180\), so \(\angle S = 180 - 88 = 92\), but \(\angle U\) is \(98\), which would mean \(\angle S eq \angle U\), which is a contradiction. So maybe my initial assumption is wrong. Wait, maybe the quadrilateral is a parallelogram (since opposite sides are equal), so it's a parallelogram (rhombus is a type of parallelogram). In a parallelogram, opposite angles are equal, adjacent angles are supplementary. So \(\angle V = \angle T\), \(\angle S = \angle U\). Then \(\angle V + \angle S = 180\), \(\angle T + \angle U = 180\). We have \(\angle U = 98\), so \(\angle S = 98\), then \(\angle V = 180 - 98 = 82\), but we also have \(\angle V = 8x\), \(\angle T = 7x + 11\), so \(8x = 7x + 11\) gives \(x = 11\), \(\angle V = 88\), which is not 82. So there's a mistake here. Wait, maybe the angle at \(U\) is not \(98^\circ\), but the angle at \(U\) is \(98^\circ\), and angles at \(V\) and \(T\) are \(8x\) and \(7x + 11\). Wait, maybe the diagram is a rhombus, so adjacent angles: \(\angle V\) and \(\angle U\) are adjacent? Wait, no, vertices in order: \(S\), \(T\), \(U\), \(V\), so sides \(ST\), \(TU\), \(UV\), \(VS\). So angles: \(\angle S\) (at \(S\) between \(ST\) and \(SV\)), \(\angle T\) (at \(T\) between \(ST\) and \(TU\)), \(\angle U\) (at \(U\) between \(TU\) and \(UV\)), \(\angle V\) (at \(V\) between \(UV\) and \(VS\)). So in a parallelogram (rhombus), \(\angle S = \angle U\), \(\angle T = \angle V\), and \(\angle S + \angle T = 180^\circ\). Ah! That's the key. So \(\angle T = \angle V\), so \(8x = 7x + 11\) (correct), so \(x = 11\), \(\angle V = 88^\circ\), \(\angle T = 88^\circ\). Then \(\angle S + \angle T = 180^\circ\) (since they are adjacent), so \(\angle S = 180 - 88 = 92^\circ\). Wait, but \(\angle U\) is \(98^\circ\), which should be equal to \(\angle S\) (opposite), but \(92 eq98\). Wait, that's a contradiction. Wait, maybe the angle at \(U\) is not \(98^\circ\), but the angle at \(U\) is \(98^\circ\), and I misread the diagram. Wait, maybe the angle at \(U\) is \(98^\circ\), and angles at \(V\) and \(T\) are \(8x\) and \(7x + 11\), and \(\angle S\) is opposite \(\angle U\), so \(\angle S = 98^\circ\), and \(\angle V\) is opposite \(\angle T\), so \(\angle V = \angle T\). Then \(\angle V + \angle S = 180^\circ\) (adjacent), so \(\angle V = 180 - 98 = 82^\circ\), so \(8x = 82\), \(x = 10.25\), but then \(7x + 11 = 7\times10.25 + 11 = 71.75 + 11 = 82.75 eq82\). So that's not equal. So there must be a mistake in my initial assumption. Wait, going back: the problem says "Find \(m\angle S\)". The diagram: quadrilateral with vertices \(S\), \(T\), \(U\), \(V\). Angles: at \(V\): \(8x\), at \(T\): \(7x + 11\), at \(U\): \(98^\circ\). Sides: \(SV\) and \(ST\) have two ticks? Wait, maybe it's a kite? No, all sides equal (rhombus). Wait, maybe the angles at \(V\) and \(T\) are equal (since it's a rhombus, opposite angles), so \(8x = 7x + 11\), so \(x = 11\), so \(\angle V = 88^\circ\), \(\angle T = 88^\circ\). Then, in a quadrilateral, sum of angles is \(360^\circ\). So \(\angle S + \angle V + \angle T + \angle U = 360\). So \(\angle S + 88 + 88 + 98 = 360\). So \(\angle S = 360 - 88 - 88 - 98 = 360 - 274 = 86\)? Wait, no, 88 + 88 is 176, 176 + 98 is 274, 360 - 274 is 86. Wait, that's different. Wait, sum of interior angles of a quadrilateral is \(360^\circ\). So that's the key! I forgot that. So sum of all angles is \(360^\circ\). So \(\angle S + \angle V + \angle T + \angle U = 360\). We know \(\angle V = 8x\), \(\angle T = 7x + 11\), \(\angle U = 98^\circ\), and \(\angle V = \angle T\) (since it's a rhombus, opposite angles), so \(8x = 7x + 11\) (so \(x = 11\)), so \(\angle V = 88\), \(\angle T = 88\). Then \(\angle S = 360 - 88 - 88 - 98 = 360 - 274 = 86\)? Wait, no, 88 + 88 is 176, 176 + 98 is 274, 360 - 274 is 86. But that contradicts the adjacent angles supplementary. Wait, in a rhombus, adjacent angles are supplementary, so \(\angle V + \angle S = 180\), so \(\angle S = 180 - 88 = 92\), and \(\angle S + \angle T = 180\), so 92 + 88 = 180 (good). Then \(\angle U\) should be equal to \(\angle S\) (opposite), so \(\angle U = 92^\circ\), but the diagram says \(\angle U = 98^\circ\). Ah! So maybe the diagram has a typo, or I misread the angle at \(U\). Wait, maybe the angle at \(U\) is \(92^\circ\), but it's written as \(98^\circ\). Alternatively, maybe my initial assumption that \(\angle V\) and \(\angle T\) are opposite is wrong. Wait, maybe the quadrilateral is a parallelogram (rhombus), so \(\angle V\) and \(\angle T\) are adjacent, not opposite. Wait, no, in a parallelogram, opposite angles are equal. So \(S\) opposite \(U\), \(V\) opposite \(T\). So \(\angle S = \angle U\), \(\angle V = \angle T\). Then sum of angles: \(2\angle S + 2\angle V = 360\), so \(\angle S + \angle V = 180\) (which is supplementary, as expected). So if \(\angle U = 98^\circ\), then \(\angle S = 98^\circ\), so \(\angle V = 180 - 98 = 82^\circ\), so \(8x = 82\), \(x = 10.25\), and \(7x + 11 = 7\times10.25 + 11 = 71.75 + 11 = 82.75 eq82\). So that's not equal. Therefore, the only way \(\angle V = \angle T\) is if \(8x = 7x + 11\), so \(x = 11\), then \(\angle V = 88\), \(\angle T = 88\), then \(\angle S = 180 - 88 = 92\), and \(\angle U\) should be 92, but it's given as 98. So maybe the angle at \(U\) is a mistake, or maybe I misread the diagram. Assuming that the angle at \(U\) is a typo and should be 92, but since it's given as 98, maybe the problem is designed so that \(\angle V = \angle T\), so solve \(8x = 7x + 11\), get \(x = 11\), then \(\angle V = 88\), then \(\angle S = 180 - 88 = 92\) (since adjacent angles in rhombus are supplementary). So despite the \(\angle U = 98\) (which might be a mistake), the correct approach is: in a rhombus, opposite angles are equal, so \(\angle V = \angle T\), solve for \(x\), then adjacent angles are supplementary, so \(\angle S = 180 - \angle V\).

So step-by-step:

1. Recognize it's a rhombus (all sides equal), so opposite angles are equal: \(\angle V = \angle T\).
2. Set \(8x = 7x + 11\), solve \(x = 11\).
3. Calculate \(\angle V = 8x = 88^\circ\).
4. In a rhombus, adjacent angles are supplementary: \(\angle S + \angle V = 180^\circ\).
5. So \(\angle S = 180 - 88 = 92^\circ\).

Answer:

\(92^\circ\)