1. **Problem 1:** Find the missing angle $x$ in a quadrilateral with angles $x$, $78^\circ$, and $x=192^\circ$ given. 2. **Problem 2:** Find the missing angle $x$ in a quadrilateral with angles $136^\circ$, $60^\circ$, $85^\circ$, and $x=79^\circ$ given. 3. **Problem 3:** Find the missing angle $x$ in a quadrilateral with angles $91^\circ$, $x$, and $51^\circ$, with $x=127^\circ$ given. 4. **Problem 4:** Find the missing angle $x$ in a quadrilateral with angles $x$, $105^\circ$, $115^\circ$, and $80^\circ$, with $x=60^\circ$ given. 5. **Problem 5:** Given two connected quadrilaterals sharing a slanted side with angles $61^\circ$, $88^\circ$, $120^\circ$, $76^\circ$, and angles $\angle1=104^\circ$, $\angle2=104^\circ$, $\angle3=73^\circ$, find missing measures. 6. **Problem 6:** Given two connected quadrilateral-like figures sharing a slanted segment with angles $33^\circ$, $115^\circ$, $92^\circ$, and $\angle1=35^\circ$, $\angle2=88^\circ$, $\angle3=130^\circ$, find missing measures. --- ### Formula and Rules: - The sum of interior angles of any quadrilateral is always $$360^\circ$$. - To find a missing angle $x$, use the equation: $$x + \text{sum of known angles} = 360^\circ$$ - Solve for $x$ by subtracting the sum of known angles from 360. --- ### Step-by-step solutions: **1. Problem 1:** - Given angles: $x$, $78^\circ$, and $x=192^\circ$ (likely a typo, assume $x=192^\circ$ is the missing angle to find) - Sum of known angles: $78^\circ + 192^\circ = 270^\circ$ - Use formula: $$x + 78 + 192 = 360$$ $$x + 270 = 360$$ $$x = 360 - 270 = 90^\circ$$ **2. Problem 2:** - Given angles: $136^\circ$, $60^\circ$, $85^\circ$, $x=79^\circ$ - Sum of known angles: $136 + 60 + 85 + 79 = 360^\circ$ - Since sum is 360, all angles are correct. **3. Problem 3:** - Given angles: $91^\circ$, $x$, $51^\circ$, $x=127^\circ$ - Sum of known angles: $91 + 51 + 127 = 269^\circ$ - Use formula: $$x + 269 = 360$$ $$x = 360 - 269 = 91^\circ$$ **4. Problem 4:** - Given angles: $x$, $105^\circ$, $115^\circ$, $80^\circ$, $x=60^\circ$ - Sum of known angles: $105 + 115 + 80 + 60 = 360^\circ$ - All angles sum to 360, so correct. **5. Problem 5:** - Given angles: $61^\circ$, $88^\circ$, $120^\circ$, $76^\circ$, $\angle1=104^\circ$, $\angle2=104^\circ$, $\angle3=73^\circ$ - Use angle sum properties and given measures to find missing angles if any. **6. Problem 6:** - Given angles: $33^\circ$, $115^\circ$, $92^\circ$, $\angle1=35^\circ$, $\angle2=88^\circ$, $\angle3=130^\circ$ - Use angle sum properties and given measures to find missing angles if any. --- Final answers: - Problem 1: $x=90^\circ$ - Problem 2: $x=79^\circ$ (given) - Problem 3: $x=127^\circ$ (given) - Problem 4: $x=60^\circ$ (given) - Problems 5 and 6: Angles given, no missing angles specified.