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.
in exercises 3 and 4, make and test a conjecture about the given quantity.
3. the sum of two negative integers
4. the product of three - consecutive nonzero integers
in exercises 5 and 6, find a counterexample to show that the conjecture is false.
5. if a is a rational number, then (a^{2}) is always less than a.
6. line a intersects plane p at point q on the plane. plane p is perpendicular to line k.
in exercises 7 and 8, use the law of detachment to determine what you can conclude from the given information, if possible.
7. if a triangle has equal side lengths, then each interior angle measure is (60^{circ}). ( riangle abc) has equal side lengths.
8. 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.
9. if it does not rain, then i will walk to school.
if i walk to school, then i will wear my sneakers.
10. if (x1), then (3x 3).
if (3x3), then ((3x)^{2}3^{2})

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.
in exercises 3 and 4, make and test a conjecture about the given quantity.
3. the sum of two negative integers
4. the product of three - consecutive nonzero integers
in exercises 5 and 6, find a counterexample to show that the conjecture is false.
5. if a is a rational number, then (a^{2}) is always less than a.
6. line a intersects plane p at point q on the plane. plane p is perpendicular to line k.
in exercises 7 and 8, use the law of detachment to determine what you can conclude from the given information, if possible.
7. if a triangle has equal side lengths, then each interior angle measure is (60^{circ}). (	riangle abc) has equal side lengths.
8. 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.
9. if it does not rain, then i will walk to school.
if i walk to school, then i will wear my sneakers.
10. if (x>1), then (3x > 3).
if (3x>3), then ((3x)^{2}>3^{2})

Accepted Answer

1. **Exercise 1**:
   - **Pattern description**: The sequence $20,19,17,14,10,\cdots$ has a pattern where the amount subtracted from each term to get the next term increases by 1 each time. First, $20 - 19=1$, then $19 - 17 = 2$, $17-14 = 3$, $14 - 10=4$.
   - **Next - two terms**:
     - To get the next term, we subtract 5 from 10. So the next term is $10-5 = 5$.
     - To get the term after that, we subtract 6 from 5. So the term is $5 - 6=-1$.
2. **Exercise 2**: The pattern is that one layer is added to the square - like figure each time. The next two figures would have one more layer added successively. (It's a visual - pattern problem, but we can't draw here, just describe the concept).
3. **Exercise 3**:
   - **Conjecture**: The sum of two negative integers is a negative integer. Let $a=-m$ and $b = -n$ where $m,n\gt0$. Then $a + b=(-m)+(-n)=-(m + n)$. Since $m + n\gt0$, $a + b$ is negative.
4. **Exercise 4**:
   - **Conjecture**: The product of three consecutive non - zero integers is a non - zero integer. Let the three consecutive non - zero integers be $n$, $n + 1$, $n+2$ where $n
eq0$. Then their product $P=n(n + 1)(n + 2)$. Since $n
eq0$, and $n+1
eq0$, $n + 2
eq0$ (because $n$ is non - zero), $P
eq0$.
5. **Exercise 5**:
   - **Counter - example**: Let $a=\frac{1}{2}$. Then $a^{2}=(\frac{1}{2})^{2}=\frac{1}{4}$, and $\frac{1}{4}\lt\frac{1}{2}$. But if $a = 2$ (a rational number), $a^{2}=4$ and $4\gt2$. So the conjecture “If $a$ is a rational number, then $a^{2}$ is always less than $a$” is false.
6. **Exercise 6**: There is no question associated with the description of the line and the plane.
7. **Exercise 7**: Given the statement “If a triangle has equal side lengths, then each interior angle measure is $60^{\circ}$” and $\triangle ABC$ has equal side lengths. By the Law of Detachment, we can conclude that each interior angle of $\triangle ABC$ measures $60^{\circ}$.
8. **Exercise 8**: Given “If a quadrilateral is a rhombus, then it has two pairs of opposite sides that are parallel” and “Quadrilateral $PQRS$ is a rhombus”. By the Law of Detachment, we can conclude that quadrilateral $PQRS$ has two pairs of opposite sides that are parallel.
9. **Exercise 9**:
   - The Law of Syllogism states that if $p
ightarrow q$ and $q
ightarrow r$, then $p
ightarrow r$. Here $p$: it does not rain, $q$: I will walk to school, $r$: I will wear my sneakers. The new conditional statement is “If it does not rain, then I will wear my sneakers”.
10. **Exercise 10**:
   - Given $p:x\gt1$, $q:3x\gt3$, $r:(3x)^{2}\gt3^{2}$. By the Law of Syllogism, the new conditional statement is “If $x\gt1$, then $(3x)^{2}\gt3^{2}$”.

# Answer:
1. Next two terms: 5, - 1
2. (Description of next two figures based on adding one layer)
3. The sum of two negative integers is a negative integer.
4. The product of three consecutive non - zero integers is a non - zero integer.
5. Counter - example: $a = 2$ (a rational number, but $a^{2}=4\gt a$)
6. No question to answer.
7. Each interior angle of $\triangle ABC$ measures $60^{\circ}$.
8. Quadrilateral $PQRS$ has two pairs of opposite sides that are parallel.
9. If it does not rain, then I will wear my sneakers.
10. If $x\gt1$, then $(3x)^{2}\gt3^{2}$

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