Determine x, y, z in a Circle with Center O

Answer

In the given circle with center $O$, the values for the unknown angles are $x = 250^\circ$, $z = 55^\circ$, and $y = 110^\circ$. These values are derived using the properties of cyclic quadrilaterals and the relationship between angles at the center and circumference.

Explanation

The image displays a circle containing a quadrilateral $ABCD$ where all vertices lie on the circumference, making it a cyclic quadrilateral. We are also given a central angle $\angle AOC$ divided into a reflex angle $x$ and an obtuse angle $y$.

1. Identifying the cyclic quadrilateral Vertices $A, B, C,$ and $D$ lie on the circle's circumference. By definition, opposite angles in a cyclic quadrilateral are supplementary (they sum to $180^\circ$). ⚠️ This step is required on exams to justify the relationship between $\angle B$ and $\angle D$. $\angle ABC + \angle ADC = 180^\circ$ This formula states that opposite internal angles of a cyclic quadrilateral always add up to a straight line's angle.

2. Calculating the value of $z$ Substitute the known value of $\angle B = 125^\circ$ into the supplementary equation. $125^\circ + z^\circ = 180^\circ$ $z = 180^\circ - 125^\circ = 55^\circ$ Subtracting the known inscribed angle from $180^\circ$ yields the opposite inscribed angle.

3. Applying the Inscribed Angle Theorem for $x$ The angle at the center is twice the angle at the circumference subtended by the same arc. Reflex angle $x$ (arc $ABC$) subtends the same arc as the inscribed angle $z$ at vertex $D$? No, look closer: reflex angle $\angle AOC$ (labeled $x$) subtends the major arc $ADC$, which is related to inscribed angle $\angle ABC$. $x = 2 \times \angle ABC$ $x = 2 \times 125^\circ = 250^\circ$ The central angle is double the inscribed angle that faces the same arc.

4. Calculating the value of $y$ Angles around a point calculate to $360^\circ$. Since $x$ and $y$ form a full circle around center $O$: $x + y = 360^\circ$ $250^\circ + y = 360^\circ$ $y = 360^\circ - 250^\circ = 110^\circ$ The sum of all angles sharing a common vertex point is $360^\circ$.

5. Verifying with the Inscribed Angle Theorem Alternatively, angle $y$ (the minor central angle $AOC$) subtends the minor arc $ABC$. Therefore, it must be twice the inscribed angle $z$ which also subtends the minor arc $ABC$. $y = 2 \times z$ $y = 2 \times 55^\circ = 110^\circ$ This confirms our previous calculation for $y$ is mathematically consistent.

Final Answer

The values of the angles are: $\boxed{x = 250, \quad y = 110, \quad z = 55}$

Common Mistakes

• Incorrect Arc Pairing: Students often mistakenly assume $y$ is double $125^\circ$. Remember that an inscribed angle is half the central angle subtended by the same arc. $125^\circ$ faces the major arc, so it relates to the reflex angle $x$.
• Supplementary Confusion: Mistaking the central angle and inscribed angle as supplementary ($y + 125 = 180$). This only applies to opposite angles on the circle's edge (circumference), not the center.