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In Exercises 11-14, find a counterexample to show that the conjecture is false. (See Example 3.) 11. The product of two positive numbers is always greater than either number. 25

In Exercises 11-14, find a counterexample to show that the conjecture is false. (See Example 3.) 11. The product of two positive numbers is always greater than either number. 25

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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**Educational Website Transcription and Description** --- ### Patterns and Sequences **Exercises 1–6:** Describe the pattern. Then write or draw the next two numbers, letters, or figures. - 1. Series: 1, –2, –3, –4, 5, … - 2. Series: 2, 0, 6, 12, 20, … - 3. Letters: X, Y, X, W, V, … - 4. Letters: A, D, G, J, M, … - 5. **Figures:** A sequence of shapes showing a pattern that starts with a triangle. - 6. **Figures:** Illustrated by cubes forming structures in a pattern. ### Conjectures **Exercises 7–10:** Make and test a conjecture about the given quantity. - 7. The sum of an even integer and an odd integer. - 8. The product of any two even integers. - 9. The quotient of a number and its reciprocal. - 10. The quotient of two negative integers. ### Counterexamples **Exercises 11–14:** Find a counterexample to show that the conjecture is false. - 11. The product of two positive numbers is always greater than either number. - 12. If \( n \) is a nonzero integer, then \( \frac{n+1}{n} \) is always greater than 1. - 13. If two angles are supplements of each other, then one of the angles must be acute. - 14. A line is divided into two line segments. So, the line is a segment bisector of \(\overline{MN}\). ### Law of Detachment **Exercises 15–18:** Use the Law of Detachment to determine what you can conclude. - 15. If you download a GIF, then your device crashes. You download a GIF. - 16. If you go to the amusement park, then you will go on the Ferris wheel. You go on the Ferris wheel. - 17. If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST has four right angles. - 18. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point \( P \) divides \(\overline{FH}\)
Transcribed Image Text:**Educational Website Transcription and Description** --- ### Patterns and Sequences **Exercises 1–6:** Describe the pattern. Then write or draw the next two numbers, letters, or figures. - 1. Series: 1, –2, –3, –4, 5, … - 2. Series: 2, 0, 6, 12, 20, … - 3. Letters: X, Y, X, W, V, … - 4. Letters: A, D, G, J, M, … - 5. **Figures:** A sequence of shapes showing a pattern that starts with a triangle. - 6. **Figures:** Illustrated by cubes forming structures in a pattern. ### Conjectures **Exercises 7–10:** Make and test a conjecture about the given quantity. - 7. The sum of an even integer and an odd integer. - 8. The product of any two even integers. - 9. The quotient of a number and its reciprocal. - 10. The quotient of two negative integers. ### Counterexamples **Exercises 11–14:** Find a counterexample to show that the conjecture is false. - 11. The product of two positive numbers is always greater than either number. - 12. If \( n \) is a nonzero integer, then \( \frac{n+1}{n} \) is always greater than 1. - 13. If two angles are supplements of each other, then one of the angles must be acute. - 14. A line is divided into two line segments. So, the line is a segment bisector of \(\overline{MN}\). ### Law of Detachment **Exercises 15–18:** Use the Law of Detachment to determine what you can conclude. - 15. If you download a GIF, then your device crashes. You download a GIF. - 16. If you go to the amusement park, then you will go on the Ferris wheel. You go on the Ferris wheel. - 17. If a quadrilateral is a square, then it has four right angles. Quadrilateral QRST has four right angles. - 18. If a point divides a line segment into two congruent line segments, then the point is a midpoint. Point \( P \) divides \(\overline{FH}\)
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