In a rhombus ABCD, the measure of angle CAB is 30°, where C, A, and B are three consecutive vertices and AB is a side. What is the measure of angle ABC (in degrees)?

Introduction / Context:
This geometry problem involves angle relationships in a rhombus and the effect of its diagonals. In a rhombus, all sides are equal, and the diagonals have special properties, including bisecting the interior angles. The question uses the angle between a diagonal and a side to determine another interior angle of the rhombus.

Given Data / Assumptions:

• ABCD is a rhombus.
• All four sides are equal in length.
• Angle CAB is 30°, where C, A, and B are consecutive vertices, and the angle is formed by CA (a diagonal) and BA (a side).
• We need to find angle ABC, which is the interior angle at vertex B.

Concept / Approach:
A key property of a rhombus is that its diagonals bisect the interior angles at each vertex. This means that each diagonal splits an interior angle into two equal angles. Here, diagonal AC passes through vertex A, bisecting angle DAB (angle at vertex A). Angle CAB is one half of angle DAB. Once we know angle A, we can find angle B using the fact that adjacent angles in a parallelogram (and hence in a rhombus) are supplementary, meaning they sum to 180°.

Step-by-Step Solution:
Step 1: Recognise that in rhombus ABCD, diagonal AC bisects angle DAB at vertex A. Step 2: Given angle CAB = 30°, note that angle CAB is one of the two equal halves of angle DAB. Step 3: Therefore, angle DAB (also called angle A) is 2 * angle CAB = 2 * 30° = 60°. Step 4: In any parallelogram, including a rhombus, adjacent interior angles are supplementary. Thus angle A + angle B = 180°. Step 5: Substitute angle A = 60°: 60° + angle B = 180°. Step 6: Solve for angle B: angle B = 180° − 60° = 120°. Step 7: Therefore, angle ABC, which is angle at vertex B, measures 120°.

Verification / Alternative check:
We can check if the other angles are consistent. In a rhombus, opposite angles are equal, so angle C also equals angle A = 60°, and angle D equals angle B = 120°. The sum of all four angles is then 60° + 120° + 60° + 120° = 360°, as required for any quadrilateral. The diagonal AC bisects angle A into two 30° angles and bisects angle C into two 30° angles as well, which is consistent with the given angle CAB = 30°.

Why Other Options Are Wrong:
An angle of 60° at B would make angle A + angle B = 60° + 60° = 120°, which contradicts the requirement that adjacent angles in a parallelogram sum to 180°. An angle of 90° at B would give angle A + angle B = 60° + 90° = 150°, also incorrect. An angle of 150° at B would make angle A + angle B = 210°, exceeding 180°. Thus 120° is the only value consistent with both the diagonal angle bisection property and the parallelogram angle sum rules.

Common Pitfalls:
Some learners mistakenly think that diagonals of a rhombus always intersect at right angles and may try to use right triangle reasoning incorrectly. Others may misinterpret angle CAB as an angle at vertex C rather than at vertex A, leading to confusion. Carefully identifying that angle CAB is at vertex A and that diagonal AC bisects angle A is crucial. Remembering that a rhombus is a special parallelogram with all sides equal but not necessarily right angles also helps frame the correct approach.

Final Answer:
The measure of angle ABC is 120°.