Question

question 4
what are the similarities of parallelograms and rhombus (focus on sides and angles)?
given each attribute, determine if it is a characteristic of parallelograms only, rhombi only, both parallelograms and rhombi, or neither parallelograms nor rhombi.

• opposite angles are congruent. select an answer
• all sides are congruent. select an answer
• consecutive angles are supplementary (add to 180°). select an answer
• all angles are congruent. select an answer
• diagonals bisect the angles. select an answer
• opposite sides are parallel. select an answer

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Explanation:

• Opposite angles are congruent: Both parallelograms and rhombuses have opposite angles that are congruent. This is a fundamental property of parallelograms (of which rhombuses are a special case).
• All sides are congruent: This is a property of rhombi only. In a general parallelogram, opposite sides are congruent, but not all four sides are necessarily congruent.
• Consecutive angles are supplementary (add to \(180^{\circ}\)): This is true for both parallelograms and rhombi. Since opposite sides are parallel (a property of parallelograms, and thus rhombi), consecutive angles are same - side interior angles, which are supplementary.
• All angles are congruent: This is a property of neither. In a parallelogram (unless it's a rectangle, which is a different special case) and a rhombus (unless it's a square, a combination of a rhombus and a rectangle), angles are not all congruent. In a parallelogram, opposite angles are congruent, and in a rhombus, opposite angles are congruent, but consecutive angles are supplementary (and usually not equal unless it's a square).
• Diagonals bisect the angles: This is a property of rhombi only. In a general parallelogram, diagonals bisect each other but do not necessarily bisect the angles.
• Opposite sides are parallel: This is a property of both parallelograms and rhombi. By definition, a parallelogram has opposite sides that are parallel, and a rhombus is a type of parallelogram.

Answer:

• Opposite angles are congruent: BOTH
• All sides are congruent: rhombi only
• Consecutive angles are supplementary (add to \(180^{\circ}\)): BOTH
• All angles are congruent: NEITHER
• Diagonals bisect the angles: rhombi only
• Opposite sides are parallel: BOTH