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Use the figure for Exercises 2-5. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid. 2. Given: AC and BD bisect each other. AC BD Conclusion: ABCD is a square.

Use the figure for Exercises 2-5. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid. 2. Given: AC and BD bisect each other. AC BD Conclusion: ABCD is a square.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.7: Distinguishable Permutations And Combinations
Problem 22E
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### Exercise Instructions **Use the figure for Exercises 2–5. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid.** **Given:** - \( \overline{AC} \) and \( \overline{BD} \) bisect each other. - \( \overline{AC} \cong \overline{BD} \) **Conclusion:** - ABCD is a square. ### Explanation of the Diagram The diagram shows a quadrilateral ABCD with diagonals \( \overline{AC} \) and \( \overline{BD} \). These diagonals intersect at a point inside the quadrilateral. ### Analysis For quadrilateral ABCD to be determined as a square based on the given conditions: 1. The diagonals \( \overline{AC} \) and \( \overline{BD} \) bisect each other. 2. \( \overline{AC} \) is congruent to \( \overline{BD} \). To conclude that ABCD is a square, the following additional information is required: - Each of the angles between the sides of the quadrilateral must be 90 degrees. - The diagonals must not only bisect each other and be of equal length, but they must also bisect at right angles. Without this information, the given conditions are insufficient to conclude that ABCD is indeed a square; they would only confirm that ABCD is a rhombus.
Transcribed Image Text:### Exercise Instructions **Use the figure for Exercises 2–5. Determine whether each conclusion is valid. If not, tell what additional information is needed to make it valid.** **Given:** - \( \overline{AC} \) and \( \overline{BD} \) bisect each other. - \( \overline{AC} \cong \overline{BD} \) **Conclusion:** - ABCD is a square. ### Explanation of the Diagram The diagram shows a quadrilateral ABCD with diagonals \( \overline{AC} \) and \( \overline{BD} \). These diagonals intersect at a point inside the quadrilateral. ### Analysis For quadrilateral ABCD to be determined as a square based on the given conditions: 1. The diagonals \( \overline{AC} \) and \( \overline{BD} \) bisect each other. 2. \( \overline{AC} \) is congruent to \( \overline{BD} \). To conclude that ABCD is a square, the following additional information is required: - Each of the angles between the sides of the quadrilateral must be 90 degrees. - The diagonals must not only bisect each other and be of equal length, but they must also bisect at right angles. Without this information, the given conditions are insufficient to conclude that ABCD is indeed a square; they would only confirm that ABCD is a rhombus.
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