Question

if two sides of a cyclic quadrilateral are parallel , prove that the other two sides are equal​

Answer 1

Step-by-step explanation:

MATHS

If two sides of a cyclic quadrilateral are parallel, prove that the remaining two sides equal and the diagonals are also equal.

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ANSWER

Let ABCD be the cyclic quadrilateral with AD parallel to BC. The center of the circle is O.

Join AC

Now, since AD is parallel to BC and AC is the transverse

angle DAC = angle ACB (interior opposite angles)

angle DAC is subtended by chord DC on the circumference and angle ACB is subtended by chord AB.

If the angles subtended by 2 chords on the circumference of the circle are equal, then the lengths of the chords is also equal.

Hence, DC = AB i.e. the 2 remaining sides of the cyclic quadrilateral are equal.

Now, BD and AC are the 2 diagonals.

In triangle ABD and triangle DCA

AB = CD (proved above)

angle DCA = angle DAB (angles subtended by the same chord AD)

angle ACB = angle DAC(proved above)

Hence, triangle ABD is congruent to triangle DCA (by ASA)

hence, BD = AC (by CPCT)

Answer 2

Answer:

Step-by-step explanation:

Draw quadrilateral ABCD, where ABllCD

Draw diagonal AC

Now you have triangles ABC and ADC

Prove congruency by SAS Test

Therefore, ADllBC

Hope it helps...:)