1. **Problem:** Find the values of $x$ and $y$ where $O$ is the center of the circle and angles satisfy $AO0 + 108 + COD + 4AO8 = 360^6$. **Step 1:** Understand that the sum of angles around point $O$ is $360^6$. **Step 2:** Given angles are $AO0 = 35^6$, $108^6$, $COD = y$, and $4AO8 = 35^6$. **Step 3:** Write the equation: $$35 + 108 + y + 35 = 360$$ **Step 4:** Simplify: $$178 + y = 360$$ **Step 5:** Solve for $y$: $$y = 360 - 178 = 182^6$$ **Step 6:** Since $x$ is not explicitly given, assume $x = 35^6$ as per the given angles. --- 2. **Problem:** In cyclic quadrilateral $ABCD$ with $AB$ as diameter and $9ADC = 140^6$, find $9BAC$. **Step 1:** Recall that angle subtended by diameter is $90^6$. **Step 2:** Use the property of cyclic quadrilateral: opposite angles sum to $180^6$. **Step 3:** Since $9ADC = 140^6$, then $9ABC = 40^6$ (because $180 - 140 = 40$). **Step 4:** Triangle $ABC$ is right angled at $C$ because $AB$ is diameter, so $9BAC + 9BCA = 90^6$. **Step 5:** Since $9BCA = 40^6$, then $$9BAC = 90 - 40 = 50^6$$ --- 3. **Problem:** Given $OA = 5$ cm, $AB = 8$ cm, and $OD ot AB$, find length $CD$. **Step 1:** Since $O$ is center and $OA = 5$ cm, $OB = 5$ cm (radius). **Step 2:** $AB = 8$ cm, so midpoint $D$ of $AB$ is at $4$ cm from $A$. **Step 3:** $OD$ is perpendicular from center to chord $AB$, so $OD$ bisects $AB$. **Step 4:** Use Pythagoras theorem in triangle $OAD$: $$OD = \sqrt{OA^2 - AD^2} = \sqrt{5^2 - 4^2} = \sqrt{25 - 16} = 3$$ cm **Step 5:** $CD$ is the length from $C$ to $D$. Without coordinates or more info, assume $C$ lies on circle such that $CD$ is chord length. More info needed to solve $CD$. --- 4. **Problem:** Two congruent circles with centers $O$ and $O'$ intersect at points $A$ and $B$. Is $9AOB = 9AO'B$ true? **Step 1:** Since circles are congruent and intersect at $A$ and $B$, arcs $AB$ in both circles are equal. **Step 2:** Angles $9AOB$ and $9AO'B$ are central angles subtending the same chord $AB$ in congruent circles. **Step 3:** Therefore, $9AOB = 9AO'B$. The statement is true. --- **Final answers:** 1. $x = 35^6$, $y = 182^6$ 2. $9BAC = 50^6$ 3. $OD = 3$ cm; $CD$ cannot be determined with given data 4. Statement is true: $9AOB = 9AO'B$