Question
opposite angles of a inscribed quadrilateral can be proven to be
Answers
GPT-4o mini
In an inscribed quadrilateral (a quadrilateral where all vertices lie on a circle), the opposite angles are supplementary. This means that the sum of the measures of each pair of opposite angles is equal to 180 degrees.
To prove this, consider a cyclic quadrilateral \(ABCD\) inscribed in a circle. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of the arc that it subtends.
1. Let \(A\) and \(C\) be the two opposite angles.
2. The angle \(A\) subtends the arc \(BC\), while angle \(C\) subtends the arc \(AB\).
3. The measure of angle \(A\) is equal to half the measure of arc \(BC\), and the measure of angle \(C\) is equal to half the measure of arc \(AB\).
4. The circles' arcs \(AB\) and \(BC\) together make a full circle, hence the entire circle measures 360 degrees.
5. Therefore, the measure of the arc \(AB + BC = 360\) degrees.
Using this, we have:
\[
\text{Measure of angle } A + \text{Measure of angle } C = \frac{1}{2}(\text{Measure of arc } BC) + \frac{1}{2}(\text{Measure of arc } AB) = \frac{1}{2}(360) = 180 \text{ degrees.}
\]
A similar argument can be applied for the other pair of opposite angles \(B\) and \(D\).
Thus, the opposite angles of an inscribed quadrilateral are supplementary, which can be summed up as:
\[
\angle A + \angle C = 180^\circ
\]
\[
\angle B + \angle D = 180^\circ
\]
To prove this, consider a cyclic quadrilateral \(ABCD\) inscribed in a circle. According to the inscribed angle theorem, the measure of an inscribed angle is half the measure of the arc that it subtends.
1. Let \(A\) and \(C\) be the two opposite angles.
2. The angle \(A\) subtends the arc \(BC\), while angle \(C\) subtends the arc \(AB\).
3. The measure of angle \(A\) is equal to half the measure of arc \(BC\), and the measure of angle \(C\) is equal to half the measure of arc \(AB\).
4. The circles' arcs \(AB\) and \(BC\) together make a full circle, hence the entire circle measures 360 degrees.
5. Therefore, the measure of the arc \(AB + BC = 360\) degrees.
Using this, we have:
\[
\text{Measure of angle } A + \text{Measure of angle } C = \frac{1}{2}(\text{Measure of arc } BC) + \frac{1}{2}(\text{Measure of arc } AB) = \frac{1}{2}(360) = 180 \text{ degrees.}
\]
A similar argument can be applied for the other pair of opposite angles \(B\) and \(D\).
Thus, the opposite angles of an inscribed quadrilateral are supplementary, which can be summed up as:
\[
\angle A + \angle C = 180^\circ
\]
\[
\angle B + \angle D = 180^\circ
\]