1. **Problem Statement:** We have two circle diagrams with angles labeled a, b, c, d, and given angles 34° and 15°. We need to find the values of these unknown angles using circle properties. 2. **Circle Properties and Formulas:** - The angle between a tangent and a chord is equal to the angle in the alternate segment. - The angle at the center of a circle is twice the angle at the circumference subtended by the same arc. - Angles in the same segment are equal. 3. **First Circle (left):** - Given angle between two radii at center O is $34^\circ$. - Angle $a$ is the angle between the tangent and the chord. - Angle $b$ is an angle at the center O. Using the tangent-chord theorem: $$a = 34^\circ$$ Since $b$ is the angle at the center subtended by the same arc as angle $a$ at the circumference, and the angle at the center is twice the angle at the circumference: $$b = 2 \times a = 2 \times 34^\circ = 68^\circ$$ 4. **Second Circle (right):** - Given angle between two chords at the circumference is $15^\circ$. - Angle $d$ is the angle at the circumference where the chords meet. - Angle $c$ is the angle at the center O. Using the property that the angle at the center is twice the angle at the circumference subtended by the same arc: $$c = 2 \times 15^\circ = 30^\circ$$ Angle $d$ is given as the angle between the chords at the circumference, so: $$d = 15^\circ$$ **Final answers:** $$a = 34^\circ, \quad b = 68^\circ, \quad c = 30^\circ, \quad d = 15^\circ$$