why inscribed quadrilaterals have opposite angles that are supplementary?
an angle inscribed in a circle is equal to half of the intercepted arc
the opposite angles of a quadrilateral intercept the entire circle
... 1/2 * 360 = 180
@Scott, I want a answer in 5-6 lines. How can I make this answer big?
Again, see:
http://www.jiskha.com/display.cgi?id=1496883964
Inscribed quadrilaterals, also known as cyclic quadrilaterals, are special types of quadrilaterals where all four vertices lie on a common circle. These quadrilaterals have some interesting properties, including the fact that opposite angles are supplementary, meaning they add up to 180 degrees.
To understand why this is true, we need to consider the properties of angles formed by intersecting lines and arcs in a circle. The most important property is the angle subtended by an arc.
When we draw two chords (line segments connecting two points on a circle) within a circle, the angles that these chords create from the center of the circle are called the angles subtended by the arcs of the circle. The key observation is that these angles are equal if and only if the chords are of equal length.
Now, let's consider an inscribed quadrilateral. Since all four vertices of the quadrilateral lie on the circle, we can draw four chords within the circle connecting adjacent vertices. The angles subtended by these chords from the center of the circle are the angles of the quadrilateral.
Now, let's focus on one pair of opposite angles of the quadrilateral. The chords corresponding to these angles are equal in length because they connect opposite vertices of the quadrilateral and these vertices lie on the same circle.
Since the chords are equal in length, the angles subtended by them from the center of the circle are also equal. Therefore, the opposite angles of an inscribed quadrilateral are equal.
Now, consider the sum of opposite angles of the inscribed quadrilateral. Since these angles are equal, their sum is twice the measure of one of the angles. This means that the sum of opposite angles is equal to 180 degrees, making them supplementary.
Therefore, in an inscribed quadrilateral, opposite angles are supplementary because the corresponding chords in the circle are equal in length, leading to equal angles subtended by those chords from the center of the circle.