Prove the theorem which states that the opposite angles of a cyclic quadrilateral are supplementary.

Question

ScanSolve AI Tutor · Accepted Answer

To prove the theorem that states the opposite angles of a cyclic quadrilateral are supplementary, follow these steps:

Draw a circle with center O and a cyclic quadrilateral ABCD inscribed in it.
A cyclic quadrilateral is a quadrilateral whose vertices all lie on a single circle.
Join the vertices B and D to the center O.
This creates two central angles: $\angle BOD$ (subtended by arc BCD) and reflex $\angle BOD$ (subtended by arc BAD).
Apply the theorem that the angle subtended by an arc at the center is twice the angle subtended by the same arc at any point on the remaining part of the circle.
For arc BCD:
$\angle BOD = 2 \angle BAD \quad \text{(Equation 1)}$
For arc BAD:
$\text{reflex } \angle BOD = 2 \angle BCD \quad \text{(Equation 2)}$
Sum the angles around the center O.
The sum of angles around a point is $360^\circ$ :
$\angle BOD + \text{reflex } \angle BOD = 360^\circ \quad \text{(Equation 3)}$
Substitute Equation 1 and Equation 2 into Equation 3.
$2 \angle BAD + 2 \angle BCD = 360^\circ$
Divide the entire equation by 2.
$\angle BAD + \angle BCD = 180^\circ$
This shows that the opposite angles $\angle A$ and $\angle C$ are supplementary.
Conclude the proof.
Similarly, by joining vertices A and C to the center O, it can be proven that $\angle ABC + \angle ADC = 180^\circ$ .
Therefore, the opposite angles of a cyclic quadrilateral are supplementary.

The proof shows that the sum of opposite angles in a cyclic quadrilateral is $\boxed{180^\circ}$ .

Answer