In triangle ABC, angle B is 35 degrees, angle C is 65 degrees, and the internal bisector of angle A meets side BC at point D. What is the measure of angle ADB, where ADB is the angle at point D in triangle ABD?

Introduction / Context:

This geometry question tests properties of triangles and angle bisectors, which are very important topics in school exams and aptitude tests. You must use the angle sum property of a triangle and then apply it carefully in a smaller triangle formed by the angle bisector.

Given Data / Assumptions:

• Triangle ABC has interior angles at vertices B and C equal to 35 degrees and 65 degrees respectively.
• The internal bisector of angle A meets side BC at point D.
• We need to find the measure of angle ADB, which is the angle at D inside triangle ABD.
• All angles are measured in degrees and ABC is a standard Euclidean triangle.

Concept / Approach:

First, use the fact that the sum of angles in a triangle is 180 degrees to find angle A. Then use the fact that AD is the internal bisector of angle A, so it splits angle A into two equal parts. Finally, work inside triangle ABD and apply the angle sum property again to find angle ADB.

Step-by-Step Solution:

Verification / Alternative check:

You can quickly check consistency by noting that angle ADB is an exterior type of angle relative to triangle ADC and is expected to be obtuse. The value 105 degrees is reasonable as it is greater than 90 degrees and consistent with the other angles in the diagram.

Why Other Options Are Wrong:

40 degrees is only half of angle A and refers to angle BAD or CAD, not ADB. The value 75 degrees is the sum of angles B and BAD, not the remaining angle. The options 90 degrees and 120 degrees do not satisfy the angle sum in triangle ABD when combined with 40 degrees and 35 degrees.

Common Pitfalls:

Learners sometimes confuse the angle at D with the angle at A, or they forget that the angle bisector applies only at vertex A. Another common error is to take 80 degrees directly as angle ADB, which is incorrect because that angle lies at vertex A, not at point D.

Final Answer:

The measure of angle ADB is 105°.