Given: quadrilateral abcd inscribed in a circle prove: ∠a and ∠c are supplementary, ∠b and ∠d are supplementary let the measure of = a°. because and form a circle, and a circle measures 360°, the measure of is 360 – a°. because of the theorem, m∠a = degrees and m∠c = degrees. the sum of the measures of angles a and c is degrees, which is equal to , or 180°. therefore, angles a and c are supplementary because their measures add up to 180°. angles b and d are supplementary because the sum of the measures of the angles in a quadrilateral is 360°. m∠a + m∠c + m∠b + m∠d = 360°, and using substitution, 180° + m∠b + m∠d = 360°, so m∠b + m∠d = 180°. what is the missing information in the paragraph proof?

Quadrilateral ABCD is inscribed in a circle. Let the measure of arc BAD be a°. Arcs BCD and BAD form a circle  and a circle measures 360°, then  measure of arc BCD is 360°-a°.

The inscribed angle theorem states that an angle  inscribed in a circle is half of the central angle  that subtends the same arc on the circle.

Because of the Inscribed angle theorem,

The sum of the measures of angles A and C is

Therefore, angles A and C are supplementary, because their measures add up to 180°.

Angles B and D are supplementary, because the sum of the measures of the angles in a quadrilateral is 360°.

m∠A + m∠C + m∠B + m∠D = 360°,

and using substitution,

180° + m∠B + m∠D = 360°, so m∠B + m∠D = 180°.

inscribed angle theorem