# The segments KE and MN intersect at point O, so the KM segment is parallel to the NE segment:

The segments KE and MN intersect at point O, so the KM segment is parallel to the NE segment: a) prove that KO ON = MO OE; b) find KM if MN = 20cm, MO = 12cm, NE = 18cm.

Since the lines KM and NE are parallel, and the lines MN and KE are secant at these lines, then

<ONE = <OMK; <OEN = <OKM as criss-cross corners.

<NOE = <KOM – vertical angles.

Three angles of one triangle are equal to three angles of another triangle, which means triangles

OKM and OEN are similar.

Therefore, we can write down the proportion:

KO: OE = MO: ON = KM: NE; that is: KO * ON = OE * MO;

KM * ON = NE * MO;

ON = MN – MO = 20 – 12 = 8 (cm);

KM = NE * MO: ON = 18 * 12: 8 = 27 (cm).

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