Question
Melvin is asked to draw a quadrilateral with the following specifications:
adjacent angles are supplementary
opposite angles are complementary
the measures of all the sides are equal
Which of the following statements about this quadrilateral is true?
A.
More than one quadrilateral exists with the given conditions, and all instances must be squares.
B.
Exactly one quadrilateral exists with the given conditions, and it must be a rhombus.
C.
No quadrilateral exists with the given conditions.
D.
More than one quadrilateral exists with the given conditions, and all instances must be rhombuses.
adjacent angles are supplementary
opposite angles are complementary
the measures of all the sides are equal
Which of the following statements about this quadrilateral is true?
A.
More than one quadrilateral exists with the given conditions, and all instances must be squares.
B.
Exactly one quadrilateral exists with the given conditions, and it must be a rhombus.
C.
No quadrilateral exists with the given conditions.
D.
More than one quadrilateral exists with the given conditions, and all instances must be rhombuses.
Answers
GPT-4o mini
To analyze the conditions specified for the quadrilateral:
1. **Adjacent angles are supplementary**: This means that each pair of adjacent angles in the quadrilateral adds up to 180 degrees.
2. **Opposite angles are complementary**: This means that each pair of opposite angles in the quadrilateral adds up to 90 degrees.
3. **All sides are equal**: This indicates that the quadrilateral is equilateral, which could mean it is a rhombus or a square.
Let's consider these conditions step-by-step:
- Since the adjacent angles are supplementary, we can label the angles of the quadrilateral as \(A\), \(B\), \(C\), and \(D\). Thus, we have:
- \(A + B = 180°\)
- \(B + C = 180°\)
- \(C + D = 180°\)
- \(D + A = 180°\)
From these equations, we can deduce \(A + C = 90°\) and \(B + D = 90°\) (which is also required by the condition of opposite angles being complementary).
- Now, we have opposite angles adding up to \(90°\):
- \(A + C = 90°\)
- \(B + D = 90°\)
Given that all angles must also satisfy the property of summing to \(360°\), we have:
- \(A + B + C + D = 360°\)
When we substitute \(C = 90° - A\) and \(D = 90° - B\) from the complementary angle conditions back into the sum:
- Substitute \(D = 90° - B\) into \(A + B + (90° - A) + (90° - B) = 360°\):
- This simplifies to \(180° = 360°\), which is a contradiction.
Thus, the conditions outlined cannot simultaneously hold true in any quadrilateral.
Therefore, the correct answer is:
**C. No quadrilateral exists with the given conditions.**
1. **Adjacent angles are supplementary**: This means that each pair of adjacent angles in the quadrilateral adds up to 180 degrees.
2. **Opposite angles are complementary**: This means that each pair of opposite angles in the quadrilateral adds up to 90 degrees.
3. **All sides are equal**: This indicates that the quadrilateral is equilateral, which could mean it is a rhombus or a square.
Let's consider these conditions step-by-step:
- Since the adjacent angles are supplementary, we can label the angles of the quadrilateral as \(A\), \(B\), \(C\), and \(D\). Thus, we have:
- \(A + B = 180°\)
- \(B + C = 180°\)
- \(C + D = 180°\)
- \(D + A = 180°\)
From these equations, we can deduce \(A + C = 90°\) and \(B + D = 90°\) (which is also required by the condition of opposite angles being complementary).
- Now, we have opposite angles adding up to \(90°\):
- \(A + C = 90°\)
- \(B + D = 90°\)
Given that all angles must also satisfy the property of summing to \(360°\), we have:
- \(A + B + C + D = 360°\)
When we substitute \(C = 90° - A\) and \(D = 90° - B\) from the complementary angle conditions back into the sum:
- Substitute \(D = 90° - B\) into \(A + B + (90° - A) + (90° - B) = 360°\):
- This simplifies to \(180° = 360°\), which is a contradiction.
Thus, the conditions outlined cannot simultaneously hold true in any quadrilateral.
Therefore, the correct answer is:
**C. No quadrilateral exists with the given conditions.**