# In a quadrilateral klmn ,kn=lm and angles knm and lmn are equal. Prove that the points k,l,m and n lie on a circle.

Grade:10

## 2 Answers

Arun
25750 Points
4 years ago
9Given: KLMN is a quadrilateral.

KN=LM, angle KNM = angle LMN.

To prove: That the quadrilateral is cyclic, i.e

All points lie on the circle.

Proof: Side KN = Side LM.......................{Given}

Therefore, opposite sides of the quadrilateral are equal.

Therefore, the quadrilateral KLMN is a Rectangle.

Now,

A quadrilateral is cyclic only when their opposite angles form 180°

But all angles of a rectangle are 90°

Therefore,

Angle KNM + Angle KLM

90° + 90°

180°

Therefore, quadrilateral KLMN is cyclic.

Therefore points K, L, M, N lie on the circle.

Aditya Gupta
2086 Points
4 years ago
dear student, note that aruns answer is totally rubbish, misleading and WRONG.
draw the fig KLMN.
join KM and LN.
now, angle KNM= angle LMN (given)
NM= MN (common side)
KN= LM (given)
by SAS, tri KNM is congruent to tri LMN.
by CPCT, KM= LN......(1)
now, KL= LK (common side)
KN= LM (given)
LN= KM (from 1)
hence by SSS, tri KNL is congruent to tri LMK.
by CPCT, angle NKL= angle MLK= x (say).
also, given angle KNM= angle LMN= y (say)
so that x+x+y+y= 360
or x+y= 180
or angle MLK + angle KNM= 180.
since the sum of opposite angles of a quad being 180 deg implies its cyclic nature (converse of cyclic quad theorem), hence we deduce that the points k,l,m and n lie on a circle.
further, note that such a quad in general shall be a trapezium, not a rectangle as arun mentioned.
KINDLY APPROVE :))

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