Question

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?

Answer 1

Answer:A

Explanation:

Inscribed angle

Answer 2

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,

`m\angle A=(a^(\circ))/(2);`

`m\angle C=(360^(\circ)-a^(\circ))/(2).`

The sum of the measures of angles A and C is

`(a^(\circ))/(2)+(360^(\circ)-a^(\circ))/(2)=180^(\circ).`

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