Question

let the measure of $overline{acd}=a^{circ}$. because $overline{acd}$ and $overline{bad}$ form a circle, and a circle measures $360^{circ}$, the measure of $overline{bad}$ is $360 - a^{circ}$. because of the theorem, $mangle a=\frac{a}{2}$ degrees and $mangle c=\frac{360 - a}{2}$ degrees. the sum of the measures of angles $a$ and $c$ is $(\frac{a}{2}+\frac{360 - a}{2})$ degrees, which is equal to $\frac{360}{2}$, or $180^{circ}$. therefore, angles $a$ and $c$ are supplementary because their measures add up to $180^{circ}$. angles $b$ and $d$ are supplementary because the sum of the measures of the angles in a quadrilateral is $360^{circ}$. $mangle a+mangle c+mangle b+mangle d = 360^{circ}$, and using substitution, $180^{circ}+mangle b+mangle d = 360^{circ}$, so $mangle b+mangle d = 180^{circ}$. what is the missing information in the paragraph proof? inscribed - angle polygon interior - angle sum quadrilateral - angle sum angle - bisector

Explanation:

Step1: Recall angle - sum property of quadrilateral

The sum of the interior angles of a quadrilateral is \(360^{\circ}\), i.e., \(m\angle A + m\angle B + m\angle C + m\angle D=360^{\circ}\).

Step2: Use supplementary - angle property

We know that angles \(B\) and \(D\) are supplementary, so \(m\angle B + m\angle D = 180^{\circ}\). Substituting this into the quadrilateral angle - sum formula: \(m\angle A+m\angle C+180^{\circ}=360^{\circ}\), then \(m\angle A + m\angle C=360^{\circ}- 180^{\circ}=180^{\circ}\).

Step3: Analyze given angle - measure relationships

Given that the measure of arc \(ACD=a^{\circ}\) and arc \(BAD\) forms a circle with arc \(ACD\), the measure of arc \(BAD = 360 - a^{\circ}\). Also, if \(m\angle A=\frac{a}{2}\) degrees and \(m\angle C=\frac{360 - a}{2}\) degrees, the sum \(m\angle A + m\angle C=\frac{a+(360 - a)}{2}=\frac{360}{2}=180^{\circ}\), which is consistent with the quadrilateral angle - sum property and supplementary - angle property. The missing information in the paragraph proof is the use of the quadrilateral interior - angle sum.

Answer:

quadrilateral interior angle sum