In trapezoid ABCD AB = BC = CD. Points K, L, M and N are the midpoints of the sides of the trapezoid.

| January 15, 2021 | education

In trapezoid ABCD AB = BC = CD. Points K, L, M and N are the midpoints of the sides of the trapezoid. Find the largest corner of the KLMN quad if the BAD angle is 40∘.

Trapezium ABCD is isosceles, since AB = CD.
Angle CBA = 180 ° – angle A = 180 ° – 40 ° = 140 ° (these are internal one-sided angles with parallel BC and AD and secant AB).
This means that the angle ВСD is also equal to 140 ° (in an isosceles trapezoid, the angles at the base are equal).
Consider triangles KBL and MCL: KB = BL = CL = CM, since points K, L and M are the midpoints of equal sides. Angle KBL = angle MCL. This means that the triangles are equal in two sides and the angle between them. Moreover, both of these triangles are isosceles.
This means that the angle BKL = BLK = CLM = CML = (180 ° – 140 °): 2 = 20 °.
The angles BLK, KLM and CLM are adjacent, which means that the angle KLM = 180 ° – (20 ° + 20 °) = 140 °.
Answer: The larger angle of the quadrilateral is 140 °.



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