Answer: To prove that AB is parallel to DC, we can use the following steps:
Draw a diagram of the cyclic quadrilateral ABCD with the point E where sides AD and BC are produced to meet.
Since ABCD is a cyclic quadrilateral, all its interior angles add up to 360 degrees. Therefore, angle A + angle B + angle C + angle D = 360 degrees.
Since AE = BE, we can say that angles A and B are equal (by corresponding angles). Similarly, angles C and D are also equal.
Since angles A and B are equal and angles C and D are also equal, we can say that angle A + angle B = angle C + angle D.
Substituting this into our equation from step 2, we get: angle A + angle B + angle C + angle D = angle A + angle B + angle A + angle B.
Simplifying, we get: 360 = 2(angle A + angle B). This means that angle A + angle B = 180.
Since angle A + angle B = 180, we can say that angle A = angle B and angle C = angle D.
Now we can use the fact that corresponding angles are equal, then angle A = angle C and angle B = angle D.
We know that opposite angles of a parallelogram are equal, so we have proved that AB is parallel to DC.
It's important to note that this proof uses the fact that quadrilateral ABCD is cyclic, which means that the sum of its angles is equal to 360 degrees. It also uses the fact that opposite angles are equal in a parallelogram, and corresponding angles are equal when two lines are parallel.
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