Question

cumulative review
match each form of reasoning with its description. 2.1, 2.4

1. deductive

a. drawing a general conclusion from specific instances

1. inductive

b. applying a general principle to a specific instance
write the symbolic notation for each form of deductive reasoning. 2.4

1. modus ponens
2. modus tollens
3. transitivity

true or false 2.7

1. if \\(\angle a\\) and \\(\angle b\\) are complementary, both \\(\angle a\\) and \\(\angle b\\) must be acute.
2. if \\(\angle a\\) and \\(\angle b\\) are supplementary, either \\(\angle a\\) or \\(\angle b\\) must be obtuse.

supply each missing statement or reason. 1.5, 2.7

1. 1. g is a point in the interior of \\(\angle def\\). 1. given
2. \\(m\angle deg + m\angle gef=m\angle def\\) 2.
3. 1. 1. given
4. \\(\angle u\\) and \\(\angle v\\) are supplementary. 2. linear pair theorem
5. 1. \\(\angle r\\) and \\(\angle s\\) are supplementary. 1. given
6. 2. definition of supplementary

Explanation:

• 43: Deductive reasoning applies a general principle to a specific instance, so 43 - b.
• 44: Inductive reasoning draws a general conclusion from specific instances, so 44 - a.
• 45: Modus ponens is symbolically represented as \(p

ightarrow q\), \(p\), then \(q\) (if \(p\) implies \(q\) and \(p\) is true, then \(q\) is true).

• 46: Modus tollens is symbolically \(p

ightarrow q\), \(
eg q\), then \(
eg p\) (if \(p\) implies \(q\) and \(q\) is false, then \(p\) is false).

• 47: Transitivity is symbolically \((p

ightarrow q)\land(q
ightarrow r)
ightarrow(p
ightarrow r)\) (if \(p\) implies \(q\) and \(q\) implies \(r\), then \(p\) implies \(r\)).

• 48: Complementary angles add up to 90 degrees. Acute angles are less than 90 degrees. So if \(\angle A\) and \(\angle B\) are complementary, both must be acute. True.
• 49: Supplementary angles add up to 180 degrees. They can be a right - angle and a right - angle (not obtuse), so the statement is false.
• 50: If a point \(G\) is in the interior of \(\angle DEF\), then by the Angle - Addition Postulate, \(m\angle DEG + m\angle GEF=m\angle DEF\).
• 51: For the Linear Pair Theorem to be applied to get that \(\angle U\) and \(\angle V\) are supplementary, the given should be that \(\angle U\) and \(\angle V\) form a linear pair.
• 52: If \(\angle R\) and \(\angle S\) are supplementary (given), by the definition of supplementary angles, \(m\angle R + m\angle S=180^{\circ}\).

Answer:

1. b. applying a general principle to a specific instance
2. a. drawing a general conclusion from specific instances
3. \(p

ightarrow q\), \(p\), then \(q\)

1. \(p

ightarrow q\), \(
eg q\), then \(
eg p\)

1. \((p

ightarrow q)\land(q
ightarrow r)
ightarrow(p
ightarrow r)\)

1. True
2. False
3. Angle - Addition Postulate
4. \(\angle U\) and \(\angle V\) form a linear pair
5. \(m\angle R + m\angle S = 180^{\circ}\)