AMSCO'S Geometry. New York - Rye High School

The Converse of the Isosceles Triangle Theorem 35712. Given: Quadrilateral ABCD with A C and BD hthe bisector ofProve:ABC.DBhbisects ADC.ABCD13. Given: AB CD, AB > CD, and AB ' BEC.Prove: AED and BEC bisect each other.ABECD14. a. Use a translation to prove that ABC and DEF in Example 2 are congruent.b. Use two line reflections to prove that ABC and DEF in Example 2 are congruent.15. Prove Corollary 9.12a, “Two right triangles are congruent if the hypotenuse and an acuteangle of one right triangle are congruent to the hypotenuse and an acute angle of the otherright triangle.”16. Prove Corollary 9.12b, “If a point lies on the bisector of an angle, it is equidistant from thesides of the angle.”17. Prove that if three angles of one triangle are congruent to the corresponding angles ofanother (AAA), the triangles may not be congruent. (Through any point on side BC ofABC, draw a line segment parallel to AC.)9-6 THE CONVERSE OF THE ISOSCELES TRIANGLE THEOREMThe Isosceles Triangle Theorem, proved in Section 5-3 of this book, is restatedhere in its conditional form. If two sides of a triangle are congruent, then the angles opposite these sidesare congruent.When we proved the Isosceles Triangle Theorem, its converse would havebeen very difficult to prove with the postulates and theorems that we had availableat that time. Now that we can prove two triangles congruent by AAS, itsconverse is relatively easy to prove.Theorem 9.13If two angles of a triangle are congruent, then the sides opposite these anglesare congruent.