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Prove the following converse, to the Vertical Angles Theorem: If A, B, C, D, and E are points such tht A * B * c, D and E are on opposite sides of doublesidearrow AB, and angle DBC approximatelyequalto angle ABE, then D, B, and E are collinear.

Solution

given angleDBC=angleABE DC andCA are joined also AE is joined now vertical angle DBC of the triangle BDC and the vertical angle ABE of the triangle BAE are equal now in triangl ABC the angle BCA is equal to angle DBC and angle BAC =angle ABE adding angle CBA on both sides we get angle DBC+angleCBA+angleABE=angleACB+angleCBA+angleBAC=180 degree[The angles of the triangle ABC] Hence D,B,E are collinear