AMSCO'S Geometry. New York - Rye High School

The Trapezoid 405Proof We will show DAB CBA. It is given that in trapezoid ABCD, AD > BC.It is given that ABCD is an isosceles trapezoid. In an isosceles trapezoid, baseangles are congruent, so DAB CBA. By the reflexive property, AB > AB.Therefore, DAB CBA by SAS. Corresponding parts of congruent trianglesare congruent, so AC > BD.Theorem 10.21bIf the diagonals of a trapezoid are congruent, then the trapezoid is isosceles.Given Trapezoid ABCD with AB CD and AC > BDDCProveAD > BCStrategy Draw DE ' AB and CF ' AB. First prove thatDEB and CFA are congruent by HL. Therefore,A E FCAB DBA. Now, prove that ACB BDA by SAS. Then AD andBC are congruent corresponding parts of congruent triangles.BThe proof of this theorem is left to the student. (See exercise 17.) Theorems10.21a and 10.21b can also be written as a biconditional.Theorem 10.21A trapezoid is isosceles if and only if the diagonals are congruent.Recall that the median of a triangle is a line segmentfrom a vertex to the midpoint of the opposite sides. Atriangle has three medians. A trapezoid has only onemedian, and it joins two midpoints.ADmedianDEFINITIONThe median of a trapezoid is a line segment whose endpoints are the midpointsof the nonparallel sides of the trapezoid.CBWe can prove two theorems about the median of a trapezoid.Theorem 10.22The median of a trapezoid is parallel to the bases.Given Trapezoid ABCD with AB CD, M themidpoint of AD, and N the midpoint of BCMDCNProveMN AB and MN CDAB