Let's work through the problem based on the image and given information.

We are asked to identify the type of quadrilateral ABCD, where opposite angles have the same measure.

Key Observations:

• The diagram shows a quadrilateral where opposite angles are congruent: $\angle A = \angle C$ and $\angle B = \angle D$.
• This symmetry suggests that the opposite angles of the quadrilateral are equal, and the opposite sides appear to be parallel.

Type of Quadrilateral:

• A quadrilateral with opposite angles congruent and opposite sides parallel is known as a parallelogram.

Justification:

1. Sum of Interior Angles: The sum of the interior angles of any quadrilateral is $360^\circ$.
2. Opposite Angles: Since $\angle A = \angle C$ and $\angle B = \angle D$, and opposite sides are parallel, it satisfies the properties of a parallelogram.
3. Parallel Sides: By extending one of the sides, alternate interior angles are created, which implies that the opposite sides are indeed parallel, confirming that ABCD is a parallelogram.

Completing the Sentences:

1. Sum of the Interior Angles: The sum of the interior angles of any quadrilateral is 360 degrees.
2. Congruent Angles: Since opposite angles are equal, $\angle A$ and $\angle D$ are congruent.
3. Parallel Sides: $\angle D$ and $\angle C$ are congruent because opposite angles in a parallelogram are equal.
4. Alternate Interior Angles: Extending one of the sides forms an alternate interior angle, showing the opposite sides are parallel.
5. Type of Quadrilateral: Thus, ABCD is a parallelogram.

Would you like more details on any part of this solution?

Here are five related questions:

1. What are the properties of a parallelogram that differentiate it from other quadrilaterals?
2. How can we prove that opposite sides of a parallelogram are equal?
3. What conditions must be met for a quadrilateral to be classified as a rhombus?
4. Can a parallelogram have angles that are not right angles? Why or why not?
5. What are the different types of parallelograms, and how are they classified?

Tip: Always remember that the sum of the interior angles of any polygon can be calculated by the formula $180^\circ \times (n - 2)$, where $n$ is the number of sides.