1. Problem 57: Find the measure of angle at U in kite XUVW where \(\angle W = 87^\circ\). 2. In a kite, two pairs of adjacent sides are equal, and one pair of opposite angles are equal. The sum of interior angles is always \(360^\circ\). 3. Since \(\angle W = 87^\circ\), and angles at U and W are opposite angles in kite, \(\angle U = \angle W = 87^\circ\). 4. Therefore, \(\boxed{87^\circ}\) is the measure at U. --- 1. Problem 58: Find \(\angle U\) in kite VSTU with \(\angle V = 60^\circ\) and \(\angle T = 100^\circ\). 2. In kite, adjacent angles between unequal sides are supplementary. 3. Since \(\angle V = 60^\circ\), \(\angle S = 60^\circ\) (opposite angles equal), and \(\angle T = 100^\circ\), \(\angle U = 100^\circ\). 4. So, \(\boxed{100^\circ}\) is the measure at U. --- 1. Problem 59: Find \(\angle R\) in kite RQPS with \(\angle Q = 96^\circ\) and \(\angle P = 65^\circ\). 2. Opposite angles in kite are equal, so \(\angle R = \angle P = 65^\circ\). 3. Therefore, \(\boxed{65^\circ}\) is the measure at R. --- 1. Problem 60: Find \(m\angle UJV\) in kite WVUX with diagonals WX and VU intersecting at J. 2. In kite, diagonals are perpendicular, so \(\angle UJV = 90^\circ\). 3. Hence, \(\boxed{90^\circ}\) is the measure of \(\angle UJV\). --- 1. Problem 61: Find unknown angle at T in kite UVST where angle between TU and TV is 55°. 2. The two angles at vertex T between adjacent sides TU-TV and TV-TS sum to 180° because they are adjacent angles on a straight line. 3. So, unknown angle = \(180^\circ - 55^\circ = 125^\circ\). 4. Therefore, \(\boxed{125^\circ}\) is the unknown angle at T. --- 1. Problem 62: Find side SP in kite RQPS where side QP = 9. 2. In kite, two pairs of adjacent sides are equal. 3. Since QP = 9, the matching side SP = 9. 4. So, \(\boxed{9}\) is the length of side SP. --- 1. Problem 63: Find side YV in kite XYVW where side XY = 18. 2. By kite property, matching adjacent sides are equal. 3. Therefore, YV = 18. 4. So, \(\boxed{18}\) is the length of side YV. --- 1. Problem 64: Find EW in kite FGEH with diagonals FH and GE intersecting at W, given GW = 9. 2. In kite, diagonals intersect such that one diagonal is bisected by the other. 3. Since GW = 9 and W lies on GE, EW = GW = 9. 4. Hence, \(\boxed{9}\) is the length of EW.