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, [tex]m\angle A=\dfrac{a^{\circ}}{2};[/tex][tex]m\angle C=\dfrac{360^{\circ}-a^{\circ}}{2}.[/tex]The sum of the measures of angles A and C is [tex]\dfrac{a^{\circ}}{2}+\dfrac{360^{\circ}-a^{\circ}}{2}=180^{\circ}.[/tex]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°.Answer: inscribed angle theorem