ABCD is a rectangle. All rectangles have congruent diagonals. Which of the following is valid based on deductive reasoning?
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To determine the validity of the statement using deductive reasoning, we need to analyze the given information and the logical implications.
Given:
1. ABCD is a rectangle.
2. All rectangles have congruent diagonals.
Statement:
- All quadrilaterals have congruent diagonals.
To check the validity, we can apply the following steps:
Step 1: Understand the definitions and given information.
- A rectangle is a type of quadrilateral, so all rectangles are quadrilaterals.
- The given information states that all rectangles have congruent diagonals.
Step 2: Apply deductive reasoning to check the validity.
- If all rectangles have congruent diagonals, and a rectangle is a type of quadrilateral, we can conclude that all quadrilaterals, not just rectangles, have congruent diagonals.
Step 3: Evaluate the statement based on deductive reasoning.
- The statement "All quadrilaterals have congruent diagonals" is valid based on deductive reasoning.
Explanation:
We know that a rectangle is a specific type of quadrilateral. The given information states that all rectangles have congruent diagonals. By applying deductive reasoning, we can extend this information to claim that all quadrilaterals have congruent diagonals. This is because any quadrilateral that is a rectangle will have congruent diagonals, and any quadrilateral that is not a rectangle cannot have congruent diagonals because it does not meet the criterion of being a rectangle.
Therefore, based on deductive reasoning, the statement "All quadrilaterals have congruent diagonals" is valid.