The problem is about a parallelogram named $DUCK$. Based on the image, $\angle DUC$ is given as $100^\circ$, and $\angle UCK$ is $75^\circ$.

Step 1: Understand the Properties of a Parallelogram

• In any parallelogram, opposite angles are equal, and adjacent angles are supplementary (i.e., their sum is $180^\circ$).

Step 2: Analyzing the Problem

Given:

• $\angle DUC = 100^\circ$
• $\angle UCK = 75^\circ$

We know that $\angle DUC$ and $\angle UCK$ form a triangle $\triangle DUC$ (since $\overline{DC}$ is a diagonal of parallelogram $DUCK$).

Step 3: Verify the Angles in Triangle $DUC$

The angles in any triangle must sum up to $180^\circ$. So, the angle $\angle DUK$ can be calculated as: $\angle DUK = 180^\circ - 100^\circ - 75^\circ = 5^\circ$

Step 4: Check for Errors

Paula says that Benji made an error. Let's verify:

If the angle $\angle DUK$ is $5^\circ$, which is implausibly small for a parallelogram given the typical size of angles in such shapes. It suggests an error in Benji's approach to determining the angles.

Conclusion:

Benji likely did not account correctly for the properties of parallelograms, specifically that the diagonal splits the parallelogram into two congruent triangles. Instead, he seems to have mistakenly calculated based on wrong assumptions about the relationship between the angles.

Would you like further details or explanations on this problem?

Here are some related questions you might consider:

1. What are the properties of the angles in a parallelogram?
2. How can we calculate the angles formed by the diagonals of a parallelogram?
3. What is the relationship between opposite angles in a parallelogram?
4. What are the implications of having such a small angle in a triangle formed within a parallelogram?
5. How do the interior angles of a triangle sum up?

Tip: When dealing with geometric figures, always remember the fundamental properties (like sum of angles in a triangle or parallelogram) as these can be very helpful in solving problems correctly.