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Triangles \(ABC\) and \(DCB\) are congruent, as shown below, where \(AB \neq AC.\)

https://latex.artofproblemsolving.com/c/e/0/ce0bd0c8c119e8879323319f2bd1f3d606500f59.png   <-----link to the image (sorry if it's super dark)

(a) Show that triangles \(ABD\) and \(DCA\) are congruent.

(b) Show that \(\angle ADB = \angle DBC.\)

Dec 20, 2019

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(a)  Assuming  that  we have an isosceles trapezoid

AB  = DC               sides of an isoceles trapezoid are equal

Angle ABC  = Angle DCB      base angles of an isoceles trapezoid are equal

Angle ABC + Angle DAB  = 180          opposite angles in an isoceles trapezoid are  supplementary

Angle  DCB  + Angle ADC  = 180

Angle DAB  = Ange  ADC                subtraction property of equality

Triangle  ABD  is congruent to Triangle DCA         By   SAS

(b)

AD  is parallel to BC            bases of anisoscees trapezoid are  parallel

Angle ADB  = Angle DBC         a transversal  cutting parallel lines   makes alternate interior angles equal   Dec 20, 2019