A circle is inscribed with quadrilateral A B C D.
Let the measure of Arc B C D = a°. Because Arc B C D and Arc B A D form a circle, and a circle measures 360°, the measure of Arc B A D is 360 – a°. Because of the ________ theorem, m∠A = StartFraction a Over 2 EndFraction degrees and m∠C = StartFraction 360 minus a Over 2 EndFraction degrees. The sum of the measures of angles A and C is (StartFraction a Over 2 EndFraction) + StartFraction 360 minus a Over 2 EndFraction degrees, which is equal to StartFraction 360 degrees Over 2 EndFraction, 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?

inscribed angle
polygon interior angle sum
quadrilateral angle sum
angle bisector

Because the angles are inscribed in the circle, the angle lie on arcs which mean that the angles have to add up to 360 degrees just like a circle is 360 degrees, making it a quadrilateral that is inscribed!

Hope this helps actually explain the answer,

Matthew Keister aka Mattsawesome 5000 YT

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PallaviB PallaviB · Answer 2 · 2022-06-23T15:04:38+02:00

The missing information in the paragraph proof is inscribed angle theorem that if an angle is inscribed in a circle, the measure of the inscribed angle is half the measure of the intercepted arc.

What is inscribed angle ?

Inscribed angle in a circle is formed by two chords that have a common end point on the circle. This common end point is the vertex of the angle.

We have,

Quadrilateral [tex]A B C D[/tex] inscribed in Circle.

Measure of Arc [tex]B C D = a^0[/tex].

Arc [tex]B C D[/tex] and Arc [tex]B A D[/tex] form a circle, and a circle measures [tex]360^0[/tex],

Measure of Arc [tex]B A D = 360 - a^0[/tex]

Because of the theorem that if an angle is inscribed in a circle, the measure of the inscribed angle is half the measure of the intercepted arc.

Therefore,

[tex]\angle A = \frac{a}{2}^0[/tex] and

[tex]\angle C = (\frac{360-a}{2}) ^0[/tex]

The sum of the measures of angles [tex]A[/tex] and [tex]C[/tex] is,

[tex]\angle A + \angle C = \frac{a}{2}^0 +(\frac{360-a}{2}) ^0=180^0[/tex]

Therefore, angles [tex]A[/tex] and [tex]C[/tex] are supplementary because their measures add up to [tex]180^0[/tex].

Angles [tex]B[/tex] and [tex]D[/tex] are supplementary because the sum of the measures of the angles in a quadrilateral is[tex]360^0[/tex].

[tex]\angle A + \angle C + \angle B + \angle D = 360^0[/tex]

and using substitution,

[tex]180^0 + \angle B + \angle D = 360^0[/tex]

So,

[tex]\angle B + \angle D = 180^0[/tex]

So, from the above provided proof we can say that the missing information in the proof was inscribed angle theorem.

Hence, we can say that the missing information in the paragraph proof is inscribed angle theorem that if an angle is inscribed in a circle, the measure of the inscribed angle is half the measure of the intercepted arc.

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