Question

points a, b, c, and d lie on circle m. line segment bd is a diameter. the measure of arc cd equals the measure of arc da. what is the measure of angle adm? 45.0° 22.5° 67.5° 30.0°

Explanation:

Step1: Find the measure of arc \( CA \)

Since \( BD \) is a diameter, the measure of arc \( BAD \) is \( 180^{\circ} \). Given that arc \( CB = \) arc \( BA \), let arc \( CB=\) arc \( BA = x\). Also, arc \( CD=\) arc \( DA = y\). Then \( 2x + 2y=180^{\circ}\), so \( x + y = 90^{\circ}\).

Step2: Use the property of central - angle and inscribed - angle relationship

The central angle \( \angle CMA=90^{\circ}\) (because \( x + y = 90^{\circ}\) and \( \angle CMA\) is the sum of the central angles corresponding to arc \( CB\) and arc \( BA\)).
Since arc \( CD=\) arc \( DA \), the central angle \( \angle CMD=\angle DMA\). And \( \angle CMA=\angle CMD+\angle DMA = 90^{\circ}\), so \( \angle DMA = 45^{\circ}\).
In \( \triangle AMD\), \( MA = MD\) (radii of the circle). Let \( \angle ADM=\angle DAM\). Using the angle - sum property of a triangle (\( \angle ADM+\angle DAM+\angle AMD = 180^{\circ}\)), and \( \angle AMD = 135^{\circ}\) (since \( \angle DMA = 45^{\circ}\) and \( \angle AMD = 180^{\circ}-\angle DMA\)).
Let \( \angle ADM = z\), then \( 2z+135^{\circ}=180^{\circ}\), \( 2z=45^{\circ}\), \( z = 22.5^{\circ}\).

Answer:

\( 22.5^{\circ}\)