1. **Problem Statement:** Find the measures of angles (a) $\angle ADX$, (b) $\angle ABC$, and (c) $\angle BCD$ given the circle with points $A$, $B$, $C$ on the circumference, points $D$ and $X$ on a horizontal line, and the angles $54^\circ$ at $D$ between $DC$ and $DX$, and $30^\circ$ at $X$ between $XD$ and $XA$. Also, $AD = BD$. 2. **Given Information and Setup:** - $AD = BD$ means triangle $ABD$ is isosceles with $AD = BD$. - $\angle D = 54^\circ$ between $DC$ and $DX$. - $\angle X = 30^\circ$ between $XD$ and $XA$. 3. **Step (a) Find $\angle ADX$:** - Since $AD = BD$, triangle $ABD$ is isosceles with $AD = BD$. - $\angle ADX$ is the angle at $D$ between $AD$ and $DX$. - Given $\angle DCX = 54^\circ$ at $D$ between $DC$ and $DX$, and $AD$ lies on the same side as $DC$ or $BD$. - Because $AD = BD$, and $BD$ is adjacent to $DC$, $\angle ADX = 54^\circ$. 4. **Step (b) Find $\angle ABC$:** - $A$, $B$, and $C$ lie on the circle. - $\angle ABC$ is an inscribed angle subtending arc $AC$. - The measure of an inscribed angle is half the measure of its intercepted arc. - To find $\angle ABC$, we need the arc $AC$. - Since $AD = BD$, triangle $ABD$ is isosceles, so $\angle BAD = \angle BDA$. - Using the given angles and circle properties, $\angle ABC = 30^\circ$ (equal to the angle at $X$). 5. **Step (c) Find $\angle BCD$:** - $\angle BCD$ is the angle at $C$ between points $B$ and $D$. - Since $D$ lies outside the circle and $C$ is on the circle, $\angle BCD$ is an angle formed by a chord and a tangent or secant. - Using the given $54^\circ$ angle at $D$ and circle properties, $\angle BCD = 54^\circ$. **Final answers:** - (a) $\angle ADX = 54^\circ$ - (b) $\angle ABC = 30^\circ$ - (c) $\angle BCD = 54^\circ$