1. **Problem statement:** In triangles \(\triangle JLK\) and \(\triangle DEF\), given \(JK = 5\), \(m\angle JKL = 60^\circ\), \(DE = 2x - 3y\), \(m\angle DEF = x + y\), and \(KL = EF\), find \(x\) and \(y\) such that \(\triangle JLK \cong \triangle DEF\). 2. **Formula and rules:** For two triangles to be congruent, corresponding sides and angles must be equal. Here, \(JK = DE\), \(m\angle JKL = m\angle DEF\), and \(KL = EF\). 3. **Set up equations:** \[ JK = DE \implies 5 = 2x - 3y \] \[ m\angle JKL = m\angle DEF \implies 60 = x + y \] 4. **Solve the system:** From the second equation, express \(x = 60 - y\). Substitute into the first: \[ 5 = 2(60 - y) - 3y = 120 - 2y - 3y = 120 - 5y \] 5. **Isolate \(y\):** \[ 5y = 120 - 5 \] \[ 5y = 115 \] \[ y = \frac{115}{5} = 23 \] 6. **Find \(x\):** \[ x = 60 - 23 = 37 \] 7. **Check:** \[ DE = 2(37) - 3(23) = 74 - 69 = 5 = JK \] \[ m\angle DEF = 37 + 23 = 60^\circ = m\angle JKL \] Thus, \(x = 37\) and \(y = 23\). --- 1. **Problem statement:** Find the measures of \(\angle A\), \(\angle B\), and \(\angle D\) in trapezoid \(ABCD\) where \(\angle D = 93^\circ\) and sides \(AD\) and \(BC\) are vertical. 2. **Properties:** In trapezoid \(ABCD\), \(AD \parallel BC\) are vertical sides, so \(AB\) and \(DC\) are horizontal. 3. **Right angles:** Since \(AD\) and \(BC\) are vertical, \(\angle A\) and \(\angle B\) are right angles: \[ \angle A = 90^\circ, \quad \angle B = 90^\circ \] 4. **Find \(\angle D\):** Given \(\angle D = 93^\circ\). 5. **Find \(\angle C\):** Sum of interior angles in quadrilateral is \(360^\circ\), so \[ \angle A + \angle B + \angle C + \angle D = 360^\circ \] \[ 90 + 90 + \angle C + 93 = 360 \] \[ \angle C = 360 - 273 = 87^\circ \] Final angles: \[ \angle A = 90^\circ, \quad \angle B = 90^\circ, \quad \angle D = 93^\circ \] --- 1. **Problem statement:** In a circle with center \(O\), points \(R\) and \(S\) on the circumference form an arc with measure \(33^\circ\). Find the value of \(x\), the angle between segments \(Q\) and \(S\) inside the circle. 2. **Rule:** The measure of an inscribed angle is half the measure of its intercepted arc. 3. **Apply rule:** If \(x\) intercepts the arc of \(33^\circ\), then \[ x = \frac{33}{2} = 16.5^\circ \] **Final answer:** \(x = 16.5^\circ\).