What can you say about any two consecutive angles in a parallelogram? a. They are always congruent. b. They are always supplementary. c. They are sometimes complementary. d. They are both right angles.
The Correct Answer and Explanation is:
The correct answer is b. They are always supplementary.
In a parallelogram, consecutive (or adjacent) angles always add up to 180 degrees, meaning they are supplementary. This property is a direct consequence of the geometric definition of a parallelogram and the parallel lines that form its sides.
To understand why this happens, consider the following:
- Parallel Lines and Transversal: In a parallelogram, opposite sides are parallel to each other. When two adjacent sides meet at a vertex, they form consecutive angles. These sides act as parallel lines, and the other sides of the parallelogram serve as a transversal that intersects them.
- Consecutive Interior Angles: According to the Consecutive Interior Angles Theorem (also known as the co-interior angles theorem), when two parallel lines are cut by a transversal, the consecutive interior angles are always supplementary. This means the sum of two consecutive angles on the same side of the transversal equals 180 degrees.
- Example: If you have a parallelogram, let’s say the angles at two adjacent vertices are labeled as angle A and angle B. Angle A and angle B are consecutive angles, and based on the property mentioned, angle A + angle B = 180 degrees. This confirms that consecutive angles are supplementary.
Thus, the property of consecutive angles always being supplementary applies to all parallelograms, whether they are rectangles, rhombuses, or general parallelograms. It is a foundational result in geometry.
However, they are not always congruent (option a), as the angles in a parallelogram can vary unless it is a special case like a rectangle or a rhombus. They are also not complementary (option c) because complementary angles sum to 90 degrees, not 180 degrees. Finally, they are not necessarily right angles (option d), as right angles only occur in rectangles or squares.
