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 \(\angle ADB = \angle DBC.\)
(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
AD = AD reflexive property
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