# What is the relationship of the opposite angles of an inscribed quadrilateral?

## What is the relationship of the opposite angles of an inscribed quadrilateral?

Conjecture (Quadrilateral Sum ): Opposite angles in any quadrilateral inscribed in a circle are supplements of each other. (Their measures add up to 180 degrees.)

## Why are opposite angles in a quadrilateral supplementary?

The angle subtended by an arc on the circle is half the measure of the arc. In a cyclic quadrilateral and which are opposite angles, are subtended by arc and arc whose combined measure is , as their sum is equal to the circumference of the circle. Hence the opposite angles of a cyclic quadrilateral are supplementary.

How do you prove a quadrilateral theorem is inscribed?

Proof: In the quadrilateral ABCD can be inscribed in a circle, then we have seen above using the inscribed angle theorem that the sum of either pair of opposite angles = (1/2(a1 + a2 + a3 + a4) = (1/2)360 = 180. Conversely, if the quadrilateral cannot be inscribed, this means that D is not on the circumcircle of ABC.

### What is the relationship of the opposite angles of a cyclic quadrilateral?

The opposite angles in a cyclic quadrilateral are supplementary. i.e., the sum of the opposite angles is equal to 180˚.

### Which quadrilaterals have opposite angles that are supplementary?

If a quadrilateral is a parallelogram, then consecutive angles are supplementary. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

What is opposite angles of a quadrilateral?

The opposite angles in a quadrilateral are those angles that are located diagonally opposite to each other. In other words, they are the angles that are connected through diagonals. For example, in the following parallelogram ABCD, ∠A and ∠C are called opposite angles. Similarly, ∠B and ∠D are opposite angles.

#### Do all quadrilaterals have opposite angles that are supplementary?

If a quadrilateral is a parallelogram, then its opposite sides are congruent. If a quadrilateral is a parallelogram, then its opposite angles are congruent. If a quadrilateral is a parallelogram, then its diagonals bisect each other. If a quadrilateral is a parallelogram, then consecutive angles are supplementary.