Question

2.2 extra practice
in exercises 1 and 2, describe the pattern. then write or draw the next two numbers or figures.

1. 20, 19, 17, 14, 10, ...
2. figure description: boxes with layers added each time

in exercises 3 and 4, make and test a conjecture about the given quantity.

1. the sum of two negative integers
2. the product of three consecutive nonzero integers

in exercises 5 and 6, find a counter - example to show that the conjecture is false.

1. if n is a rational number, then n² is always less than n.
2. line l intersects plane p at point q on the plane. plane p is perpendicular to line l.

in exercises 7 and 8, use the law of detachment to determine what you can conclude from the given information, if possible.

1. if a triangle has equal side lengths, then each interior angle measure is 60°. △abc has equal side lengths.
2. if a quadrilateral is a rhombus, then it has two pairs of opposite sides that are parallel. quadrilateral pqrs has two pairs of opposite sides that are parallel.

in exercises 9 and 10, use the law of syllogism to write a new conditional statement that follows from the pair of true statements, if possible.

1. if it does not rain, then i will walk to school. if i walk to school, then i will wear my sneakers.
2. if x > 1, then 3x > 3. if 3x > 3, then (3x)² > 3².

Explanation:

Step1: Analyze problem 1

The sequence 20, 19, 17, 14, 10 subtracts 1, then 2, then 3, then 4 successively. The next subtraction should be 5, so 10 - 5=5, and then subtracting 6 gives 5 - 6=-1.

Step2: Analyze problem 2

The pattern is adding one layer each time. The next two figures would have more layers added following the pattern.

Step3: Analyze problem 3

Let the two negative - integers be -a and -b where a > 0 and b > 0. Then (-a)+(-b)=-(a + b), which is a negative integer.

Step4: Analyze problem 4

Let the three consecutive non - zero integers be n, n + 1, n + 2. Their product is n(n + 1)(n + 2). Since n≠0, n+1≠0, and n + 2≠0, the product is non - zero.

Step5: Analyze problem 5

A counter - example for the conjecture "If n is a rational number, then n² is always less than n" is n = 2. Since 2²=4 and 4>2.

Step6: Analyze problem 6

A counter - example is when the line intersects the plane at a point but is not perpendicular. For example, a line can slant through the plane at a non - right angle at the intersection point.

Step7: Analyze problem 7

By the Law of Detachment, since the hypothesis "A triangle has equal side lengths" is true for △ABC and the conditional statement "If a triangle has equal side lengths, then each interior angle measure is 60°" is given, we can conclude that each interior angle of △ABC measures 60°.

Step8: Analyze problem 8

The Law of Detachment cannot be applied here. Just because a quadrilateral PQRS has two pairs of opposite sides that are parallel does not mean it is a rhombus. The conditional statement has the form "If a quadrilateral is a rhombus, then it has two pairs of opposite sides that are parallel", and we are given the conclusion of the conditional statement as true for PQRS, but we cannot infer the hypothesis.

Step9: Analyze problem 9

By the Law of Syllogism, if p: it does not rain, q: I will walk to school, r: I will wear my sneakers, and we have p→q and q→r, then p→r. So the new conditional statement is "If it does not rain, then I will wear my sneakers".

Step10: Analyze problem 10

By the Law of Syllogism, if p: x > 1, q: 3x>3, r:(3x)²>3², and we have p→q and q→r, then p→r. So the new conditional statement is "If x > 1, then (3x)²>3²".

Answer:

1. The next two numbers are 5 and - 1.
2. Draw the figures with more layers following the pattern.
3. The sum of two negative integers is a negative integer.
4. The product of three consecutive non - zero integers is non - zero.
5. Counter - example: n = 2.
6. A non - perpendicular intersection example.
7. Each interior angle of △ABC measures 60°.
8. No valid conclusion using the Law of Detachment.
9. If it does not rain, then I will wear my sneakers.
10. If x > 1, then (3x)²>3².