KLMN

In the trapezoid KLMN is given this informations:
1. segments KL and MN are parallel
2. segments KL and KM has same length
3. segments KN, NM and ML has same length.

Determine the size of the angle KMN.

Correct result:

α =  36 °

Solution:

|KL| = |KM| \ \\ |KN|=|NM|=|ML| \ \\ \alpha = ∠ KMN = ∠ NKM = ∠ MKL \ \\ \beta = ∠ KNM \ \\ \ \\ ∠ KML = \beta - \alpha \ \\ ∠ KLM = 180 ^\circ - \alpha - (\beta - \alpha) = 180 ^\circ - \beta \ \\ \ \\ \ \\ Δ KLM: \ \\ ∠ KML = ∠ KLM \ \\ \beta -\alpha = 180 ^\circ - \beta \ \\ \beta = 180 ^\circ + \alpha/2; \ \\ \ \\ Δ KMN: \ \\ 2\alpha + \beta = 180^\circ \ \\ 2\alpha + 90+ \alpha /2 = 180^\circ \ \\ 2.5 \alpha = 90^\circ \ \\ \ \\ \alpha = 36 ^\circ



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