Question

given: a circle with inscribed quadrilateral abcd
prove: ∠a and ∠c are supplementary.

1. let ( moverarc{dcb} = a^circ )

then ( moverarc{dab} = 360 - a ).

1. by the inscribed angle theorem, ( mangle a = a/2 ).
2. also by the inscribed angle theorem, ( mangle c = (360 - a)/2 ).
3. therefore, ( mangle a + mangle c = \frac{a}{2} + \frac{360 - a}{2} ),

which simplifies to 180.

1. therefore, ( mangle a ) and ( mangle c ) are dropdown with options: complementary, supplementary by the

definition of supplementary angles.
(image of circle with inscribed quadrilateral abcd: vertices a, b, c, d on the circle, center marked.)

Explanation:

To determine the relationship between \( m\angle A \) and \( m\angle C \), we use the definition of supplementary angles: two angles are supplementary if their measures add up to \( 180^\circ \). From step 4, we found that \( m\angle A + m\angle C = 180^\circ \). By the definition of supplementary angles, this means \( \angle A \) and \( \angle C \) are supplementary.

Answer:

supplementary