Question

which additional given information is sufficient for showing that triangle dbc is isosceles? select all that apply. angle abd is congruent to angle abc. angle bad is congruent to angle abc. triangle dab is congruent to triangle cab. angle bdc is congruent to angle bcd. line ab is an angle bisector of dac.

Explanation:

Step1: Recall isosceles - triangle definition

An isosceles triangle has two equal - length sides or two equal - measure angles.

Step2: Analyze each option

• Option 1: Angle \(ABD\) is congruent to angle \(ABC\). This information is about angles outside of \(\triangle DBC\) and does not directly imply that \(\triangle DBC\) is isosceles.
• Option 2: Angle \(BAD\) is congruent to angle \(ABC\). This information is also about angles not directly related to the sides or angles of \(\triangle DBC\) for showing it is isosceles.
• Option 3: If \(\triangle DAB\cong\triangle CAB\), then \(DB = CB\) by the property of congruent triangles (corresponding parts of congruent triangles are congruent). So, \(\triangle DBC\) is isosceles.
• Option 4: If \(\angle BDC\cong\angle BCD\), then by the isosceles - triangle theorem (if two angles of a triangle are congruent, then the sides opposite those angles are congruent), \(DB = CB\) and \(\triangle DBC\) is isosceles.
• Option 5: Line \(AB\) is an angle - bisector of \(\angle DAC\). This information is about \(\angle DAC\) and does not directly show that \(\triangle DBC\) is isosceles.

Answer:

C. Triangle \(DAB\) is congruent to triangle \(CAB\).
D. Angle \(BDC\) is congruent to angle \(BCD\).