Question

which arc is intercepted by ∠a in quadrilateral abcd inscribed in a circle? a. arc ab b. arc ad c. arc bc d. arc bcd what is the sum of opposite angles in any quadrilateral inscribed in a circle? a. 360 degrees b. 270 degrees c. 180 degrees d. 90 degrees what is the sum of all angles in any quadrilateral? a. 360 degrees b. 270 degrees c. 180 degrees d. 120 degrees

Explanation:

Step1: Recall inscribed - angle theorem

An inscribed angle is half of the central angle that subtends the same arc. For $\angle A$ in cyclic quadrilateral $ABCD$, the arc it intercepts is arc $BCD$.

Step2: Recall property of cyclic quadrilaterals

The sum of opposite angles in a cyclic quadrilateral is $180^{\circ}$ because the sum of the measures of the arcs of a circle is $360^{\circ}$ and the inscribed - angle relationship.

Step3: Recall angle - sum property of quadrilaterals

The sum of the interior angles of any quadrilateral is $360^{\circ}$ by the formula $(n - 2)\times180^{\circ}$ where $n = 4$ for a quadrilateral.

Answer:

1. d. Arc $BCD$
2. c. $180$ degrees
3. a. $360$ degrees