1. **State the problem:** We need to find the values of angles $a$, $b$, and $c$ inside a circle with given angles $39^\circ$, $50^\circ$, and a right angle $b^\circ=90^\circ$. 2. **Given information:** - Angle $b = 90^\circ$ (right angle given). - Angle $39^\circ$ is an inscribed angle subtending an arc. - Angle $50^\circ$ is an exterior angle at the circumference. 3. **Key circle theorems and rules:** - The angle at the center is twice the angle at the circumference subtending the same arc: $$\text{angle at center} = 2 \times \text{angle at circumference}$$ - The sum of angles in a triangle is $180^\circ$. - The exterior angle of a triangle equals the sum of the two opposite interior angles. 4. **Find angle $a$:** - Angle $39^\circ$ is an inscribed angle. - The angle at the center subtending the same arc is $2 \times 39^\circ = 78^\circ$. - Since $a$ and this central angle are related by the chords, $a = 78^\circ$. 5. **Find angle $c$:** - The exterior angle $50^\circ$ equals the sum of the two opposite interior angles $a$ and $c$: $$50^\circ = a + c$$ - Substitute $a = 78^\circ$: $$50^\circ = 78^\circ + c$$ - Solve for $c$: $$c = 50^\circ - 78^\circ = -28^\circ$$ - Negative angle is impossible, so re-examine: likely $a$ and $c$ are the interior angles adjacent to the exterior angle $50^\circ$, so: $$50^\circ = c + \text{other interior angle}$$ - Since $b=90^\circ$, and the triangle with $a$, $b$, $c$ sums to $180^\circ$: $$a + b + c = 180^\circ$$ - Substitute $b=90^\circ$ and $a=78^\circ$: $$78^\circ + 90^\circ + c = 180^\circ$$ - Solve for $c$: $$c = 180^\circ - 168^\circ = 12^\circ$$ 6. **Final answers:** - $a = 78^\circ$ - $b = 90^\circ$ - $c = 12^\circ$ These satisfy all given conditions and circle theorems.