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In circle X shown, mAS^(⏜)=11^(@) and mMS^(⏜)=104^(@). Determine m/_DCM.

44. In circle X X shown, m \overparen{A S}=11^{\circ} and m \overparen{M S}=104^{\circ} . Determine m∠DCM m \angle D C M .

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Q. 44. In circle X X shown, m \overparen{A S}=11^{\circ} and m \overparen{M S}=104^{\circ} . Determine m∠DCM m \angle D C M .
  1. Identify Relationship: Identify the relationship between the arcs and the angles in a circle. In a circle, the measure of an angle formed by two chords that intersect inside the circle is equal to half the sum of the measures of the arcs intercepted by the angle and its vertical angle.
  2. Calculate Angle ASM: Calculate the measure of angle ASM using the given arc measures.\newlineThe measure of angle ASM is half the sum of the measures of arcs AS and MS.\newlinem∠ASM=mAS◯+mMS◯2m\angle ASM = \frac{mAS^{\bigcirc} + mMS^{\bigcirc}}{2}\newlinem∠ASM=11∘+104∘2m\angle ASM = \frac{11^\circ + 104^\circ}{2}\newlinem∠ASM=115∘2m\angle ASM = \frac{115^\circ}{2}\newlinem∠ASM=57.5∘m\angle ASM = 57.5^\circ
  3. Determine Angle DCM: Determine the measure of angle DCM.\newlineSince angle DCM is vertical to angle ASM, they are congruent and have the same measure.\newlinem∠DCM=m∠ASMm\angle DCM = m\angle ASM\newlinem∠DCM=57.5∘m\angle DCM = 57.5^\circ

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