AMSCO'S Geometry. New York - Rye High School

404 Quadrilaterals(Continued)StatementsReasonsTS7. SPR R 7. If two sides of a triangle arecongruent, the angles oppositethese sides are congruent.QPR8. Q SPR 8. If two parallel lines are cut by atransversal, the correspondingangles are congruent.9. Q R 9. Transitive property of congruence.We have proved Theorem 10.20a for Q R but S and T are alsocongruent base angles. We often refer to Q and R as the lower base anglesand S and T as the upper base angles. The proof of this theorem for S andT is left to the student. (See exercise 15.)Theorem 10.20bIf the base angles of a trapezoid are congruent, then the trapezoid is isosceles.Given Trapezoid QRST with QR ST and Q RTSProveQT > RSStrategy Draw SP TQ. Prove SPR R. Then use theconverse of the isosceles triangle theorem.QPRThe proof of this theorem is left to the student. (See exercise 16.) Theorems10.20a and 10.20b can be written as a biconditional.Theorem 10.20A trapezoid is isosceles if and only if the base angles are congruent.We can also prove theorems about the diagonals of an isosceles trapezoid.Theorem 10.21aIf a trapezoid is isosceles, then the diagonals are congruent.Given Isosceles trapezoid ABCD with AB CD andAD > BCDCProveAC > BDAB