In a rhombus ABCD, the diagonal AC is drawn and the measure of angle CAB is 35°. Using the properties of rhombus diagonals and interior angles, what is the measure of interior angle ABC of the rhombus?

Introduction / Context:
This question comes from Euclidean geometry and tests your understanding of the properties of a rhombus, especially how its diagonals relate to its interior angles. You are given an angle formed by a diagonal and a side and asked to find one of the interior angles of the rhombus.

Given Data / Assumptions:

• ABCD is a rhombus.
• Diagonal AC is drawn.
• Angle CAB (an angle at vertex A between CA and AB) is 35 degrees.
• All standard geometric properties of a rhombus hold: all sides are equal, and diagonals bisect angles.

Concept / Approach:
In a rhombus, each diagonal bisects the interior angle at the vertices it joins. Thus, diagonal AC cuts angle A into two equal parts. If one of those parts (angle CAB) is known, we can find the full angle at A. Then, using the fact that adjacent interior angles of a parallelogram (and hence a rhombus) are supplementary, we can find angle B.

Step-by-Step Solution:
Step 1: In rhombus ABCD, diagonal AC bisects angle A. Step 2: We are given angle CAB = 35 degrees. Step 3: Because AC bisects angle A, angle DAB and angle CAB are equal. Step 4: Therefore, angle A = angle DAB + angle CAB = 35 + 35 = 70 degrees. Step 5: A rhombus is a special parallelogram, so adjacent interior angles are supplementary. That is, angle A + angle B = 180 degrees. Step 6: Substitute angle A = 70 degrees: 70 + angle B = 180. Step 7: Solve for angle B: angle B = 180 − 70 = 110 degrees. Step 8: Thus, the measure of angle ABC is 110 degrees.

Verification / Alternative check:
Another way to think about it is that opposite angles of a rhombus are equal, and diagonals bisect these angles. If angle A is 70 degrees, then angle C is also 70 degrees. The total of all interior angles in any quadrilateral is 360 degrees. So angle B + angle D = 360 − (70 + 70) = 220 degrees. Since B and D are equal in a rhombus, each must be 110 degrees. This again confirms that angle ABC is 110 degrees.

Why Other Options Are Wrong:
70 degrees is the measure of angle A, not angle B.
40 degrees and 50 degrees do not satisfy the supplementary relationship with 70 degrees and do not fit the properties of a rhombus.
90 degrees would indicate a right angle, which is not supported by the given data because it would give each half-angle as 45 degrees, not 35 degrees.

Common Pitfalls:
A common mistake is to assume that the angle given (35 degrees) is the interior angle itself rather than half of it. Another pitfall is forgetting that a rhombus is also a parallelogram and therefore adjacent angles are supplementary. Always use both diagonal and parallelogram properties when dealing with rhombus questions.

Final Answer:
The measure of angle ABC is 110°.