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The diagonals of parallelogram WXYZ intersect at P. Which statement must be true? Select all that apply. A. WP YP B. XY WZ C. ZYWZ EZWYX D. MZWXY+ MZYZW = 180°

The diagonals of parallelogram WXYZ intersect at P. Which statement must be true? Select all that apply. A. WP YP B. XY WZ C. ZYWZ EZWYX D. MZWXY+ MZYZW = 180°

Elementary Geometry For College Students, 7e
Elementary Geometry For College Students, 7e
7th Edition
ISBN:9781337614085
Author:Alexander, Daniel C.; Koeberlein, Geralyn M.
Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.
ChapterP: Preliminary Concepts
SectionP.CT: Test
Problem 1CT
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**The Diagonals of a Parallelogram: Analysis and Properties** In a parallelogram WXYZ, the diagonals intersect at point P. Based on the properties of parallelograms and their diagonals, determining which of the following statements must be true is fundamental in understanding the geometric relationships within parallelograms. Consider the following statements. You are to select all that apply: A. \( WP = YP \) B. \( \frac{XY}{YZ} = \frac{WX}{WZ} \) C. \( \angle WZ = \angle WYX \) D. \( m \angle WXY + m \angle YZW = 180^\circ \) **Explanation of Choices:** A. **\( WP = YP \):** This statement suggests that the diagonals bisect each other. In the case of a parallelogram, this is indeed true. The diagonals of a parallelogram always bisect each other, which means WP = YP. B. **\( \frac{XY}{YZ} = \frac{WX}{WZ} \):** This statement implies a ratio equality between the sides of parallelogram WXYZ. There is no inherent property in parallelograms that states a direct proportional relationship among the lengths of adjacent and opposite sides like this, making this statement generally false. C. **\( \angle WZ = \angle WYX \):** This statement seems to imply angle congruence based on perhaps an incorrect assumption. In a parallelogram, opposite angles are equal, not angles formed at the vertices with respect to the diagonals. D. **\( m \angle WXY + m \angle YZW = 180^\circ \):** This statement posits that the sum of consecutive angles in a parallelogram is 180°. This is accurate because in a parallelogram, consecutive angles are supplementary. **Conclusion:** Based on an understanding of parallelogram properties, the correct selections are: - **A. \( WP = YP \)** - **D. \( m \angle WXY + m \angle YZW = 180^\circ \)**
Transcribed Image Text:**The Diagonals of a Parallelogram: Analysis and Properties** In a parallelogram WXYZ, the diagonals intersect at point P. Based on the properties of parallelograms and their diagonals, determining which of the following statements must be true is fundamental in understanding the geometric relationships within parallelograms. Consider the following statements. You are to select all that apply: A. \( WP = YP \) B. \( \frac{XY}{YZ} = \frac{WX}{WZ} \) C. \( \angle WZ = \angle WYX \) D. \( m \angle WXY + m \angle YZW = 180^\circ \) **Explanation of Choices:** A. **\( WP = YP \):** This statement suggests that the diagonals bisect each other. In the case of a parallelogram, this is indeed true. The diagonals of a parallelogram always bisect each other, which means WP = YP. B. **\( \frac{XY}{YZ} = \frac{WX}{WZ} \):** This statement implies a ratio equality between the sides of parallelogram WXYZ. There is no inherent property in parallelograms that states a direct proportional relationship among the lengths of adjacent and opposite sides like this, making this statement generally false. C. **\( \angle WZ = \angle WYX \):** This statement seems to imply angle congruence based on perhaps an incorrect assumption. In a parallelogram, opposite angles are equal, not angles formed at the vertices with respect to the diagonals. D. **\( m \angle WXY + m \angle YZW = 180^\circ \):** This statement posits that the sum of consecutive angles in a parallelogram is 180°. This is accurate because in a parallelogram, consecutive angles are supplementary. **Conclusion:** Based on an understanding of parallelogram properties, the correct selections are: - **A. \( WP = YP \)** - **D. \( m \angle WXY + m \angle YZW = 180^\circ \)**
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