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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.

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
25763 Points
one year 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
2080 Points
one year 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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