Question

in the accompanying diagram, (overline{bd}) bisects acute angle (abc) and (overline{ad}paralleloverline{bc}). which kind of triangle must (\triangle abd) be? scalene isosceles right equilateral

Explanation:

Step1: Use the angle - bisector property

Since \(BD\) bisects \(\angle ABC\), we have \(\angle ABD=\angle DBC\).

Step2: Use the parallel - line property (alternate interior angles)

Because \(AD\parallel BC\), by the alternate interior angles theorem, \(\angle ADB = \angle DBC\).

Step3: Equate the angles

From steps 1 and 2, \(\angle ABD=\angle ADB\).

Step4: Use the property of a triangle (equal angles imply equal sides)

In \(\triangle ABD\), if \(\angle ABD=\angle ADB\), then \(AB = AD\) (sides opposite to equal angles in a triangle are equal).

Answer:

isosceles