Question

1. triangle dac is isosceles with congruent sides ad and ac. which additional given information is sufficient for showing that triangle dbc is isosceles? select all that apply. a. line ab is an angle bisector of dac. b. angle bad is congruent to angle abc. c. angle bdc is congruent to angle bcd. d. angle abd is congruent to angle abc. e. triangle dab is congruent to triangle cab.

Explanation:

Step1: Recall isosceles - triangle properties

An isosceles triangle has two congruent sides. We know that in \(\triangle DAC\), \(AD = AC\).

Step2: Analyze option A

If \(AB\) is the angle - bisector of \(\angle DAC\), then \(\angle DAB=\angle CAB\). Also, \(AD = AC\) and \(AB = AB\) (common side). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle DAB\cong\triangle CAB\). Then \(BD = BC\), so \(\triangle DBC\) is isosceles.

Step3: Analyze option B

If \(\angle BAD=\angle ABC\), this does not directly lead to the conclusion that \(BD = BC\). There is not enough information to prove the congruence of two triangles that would give us equal sides in \(\triangle DBC\).

Step4: Analyze option C

If \(\angle BDC=\angle BCD\), then by the definition of an isosceles triangle (if two angles of a triangle are congruent, the sides opposite them are congruent), \(BD = BC\), so \(\triangle DBC\) is isosceles.

Step5: Analyze option D

If \(\angle ABD=\angle ABC\), and \(AD = AC\), \(AB = AB\), but this is not a valid congruence - making condition for the triangles in a way that gives \(BD = BC\). There is not enough information to prove \(\triangle DBC\) is isosceles.

Step6: Analyze option E

If \(\triangle DAB\cong\triangle CAB\), then by the definition of congruent triangles, \(BD = BC\). So \(\triangle DBC\) is isosceles.

Answer:

A. Line \(AB\) is an angle bisector of \(DAC\), C. Angle \(BDC\) is congruent to angle \(BCD\), E. Triangle \(DAB\) is congruent to triangle \(CAB\)