Question

1. identify the hypothesis and the conclusion. then rewrite the conditional statement in if-then form.

all ( 22^{circ} ) angles are acute angles.

1. are ( overline{eb} ) and ( overline{bd} ) opposite rays?
2. complete the truth table for hypothesis ( p ) and conclusion ( q ).
3. find a counterexample to show that the conjecture is false.

all numbers ending in 6 are divisible by 6.

1. use the law of syllogism to write a new conditional statement that follows from the pair of true statements.

if ( 2x + 5 = 15 ), then ( x = 5 ).
if ( x = 5 ), then ( -3x = -15 ).

Explanation:

1. Hypothesis and Conclusion

• Hypothesis: A \(22^{\circ}\) angle.
• Conclusion: It is an acute angle.
• If - then form: If an angle is \(22^{\circ}\), then it is an acute angle.

2. Opposite Rays

Opposite rays are two rays that have the same endpoint and form a straight line. For \(\overrightarrow{EB}\) and \(\overrightarrow{BD}\), they do not have the same endpoint.

3. Truth Table

• When \(p = T\) and \(p

ightarrow q=T\), then \(q = T\) (because \(T
ightarrow T=T\)).

• When \(p = T\) and \(p

ightarrow q = F\), then \(q = F\) (because \(T
ightarrow F=F\)).

• When \(p = F\), \(p

ightarrow q=T\) for both \(q = T\) and \(q = F\) (because \(F
ightarrow T=T\) and \(F
ightarrow F=T\)).

The completed truth table:

\(p\)

\(q\)

\(p

ightarrow q\) |

\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)
\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)

4. Counter - example

Answer:

s:

1. Hypothesis: A \(22^{\circ}\) angle; Conclusion: It is an acute angle; If - then form: If an angle is \(22^{\circ}\), then it is an acute angle.
2. No.
3. The completed truth table:

\(p\)

\(q\)

\(p

ightarrow q\) |

\(T\)	\(T\)	\(T\)
\(T\)	\(F\)	\(F\)
\(F\)	\(T\)	\(T\)
\(F\)	\(F\)	\(T\)

1. Counter - example: \(16\).
2. If \(2x + 5 = 15\), then \(-3x=-15\).