https://file.aiquickdraw.com/mathbot/file/172767023207239vuh48w.jpg

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.

ABCD is a quadrilateral in which opposite angles have the same measure, as indicated in the following figure. What kind of quadrilateral is ABCD? Justify your answer.

This problem explores how to identify a parallelogram by examining a quadrilateral with opposite angles congruent. Learn about the properties of parallelograms, the sum of interior angles, and the alternate interior angle theorem. Suitable for Grades 8-10.

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