Question

(1) The diagonals of cyclic quadrilateral ABCD are congruent. Show that AD = BC
and seg AB || seg CD.​

Answer 1

Line AB is parallel to line CD.

AD = BC

Step-by-step explanation:

First, we draw a cyclic quadrilateral and draw the diagonals. They intersect at the point O. (refer the figure attached)

• It is given that both the diagonals are congruent.

        It means that the lengths of both diagonals are equal.

        Therefore, length of OA = length of OC

                  And, length of OB = length of OD

Since, Both the diagonals are equal in length.

So, we can write as;

• length of OA = length of OC = length of OB = length of OD

Hence, we can say that $\Delta OAB, \Delta OBC, \Delta OCD, \Delta ODA$are isosceles triangles.

• Also, $\angle AOB = \angle BOC = \angle COD = \angle DOA = \dfrac {360^o}{4} = 90^o$

Thus, we can conclude that the lines of quadrilateral are equal in measures.

• So, this quadrilateral is a square. It means, AD = BC.

         Therefore, line AB is parallel to line CD.

Answer 2

Answer:

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Step-by-step explanation:

Line AB is parallel to line CD.

AD = BC

Step-by-step explanation:

First, we draw a cyclic quadrilateral and draw the diagonals. They intersect at the point O. (refer the figure attached)

It is given that both the diagonals are congruent.

It means that the lengths of both diagonals are equal.

Therefore, length of OA = length of OC

And, length of OB = length of OD

Since, Both the diagonals are equal in length.

So, we can write as;

length of OA = length of OC = length of OB = length of OD

Hence, we can say that ΔOAB, ΔOBC, ΔOCD, and ΔODA are isosceles triangles.

Also, angle AOB = angle BOC = angle COD = \

angle DOA = 360° / 4 = 90°

Thus, we can conclude that the lines of quadrilateral are equal in measures.

So, this quadrilateral is a square. It means, AD = BC.

Therefore, line AB is parallel to line CD.