Question

chapter 3: angle pairs & perpendicular lines
select the reverse of each definition below and tell whether it passes the reversibility test.

1. (2) u.s. founding fathers authors of the federalist james madison

a. if a person was an author of the federalist, then he was a u.s. founding father. if james madison was a person, then he was a u.s. founding father.
b. if a person was a u.s. founding father, then he was an author of the federalist. if james madison was a person, then he was an author of the federalist.
c. if a person was an author of the federalist, then he was a u.s. founding father. if james madison was an author of the federalist, then he was a u.s. founding father.
d. if a person was an author of the federalist, then he was a u.s. founding father. if james madison was a u.s. founding father, then he was an author of the federalist.
e. if a person was a u.s. founding father, then he was an author of the federalist. if james madison was a u.s. founding father, then he was an author of the federalist.

1. (7) a unicycle is a vehicle with a single wheel that is propelled by pedals.

a. a vehicle with a single wheel that is propelled by pedals is a unicycle; passes
b. a single wheel on a vehicle that is propelled by pedals is a unicycle; does not pass
c. a vehicle propelled by pedals is a unicycle; passes
d. a vehicle propelled by pedals is a unicycle; does not pass
e. a vehicle with a single wheel that is propelled by pedals is a unicycle; does not pass

1. (7) a department store is a place to purchase clothing.

a. a place to purchase clothing is a department store; passes
b. a place that purchases clothing is a department store; passes
c. a place to purchase clothing is a department store; does not pass
d. all clothing is purchased at department stores; does not pass
e. a place that purchases clothing is a department store; does not pass
do the proof below.
(e) 23. given: mo bisects ∠npq; ∠npo and ∠npm are supplementary; ∠opq and ∠qpm are supplementary. prove: m∠qpm = m∠npm

Explanation:

Step1: Analyze question 20

The set - relationship is that authors of The Federalist are a subset of U.S. Founding Fathers and James Madison is an author of The Federalist. The correct logical statement is: If a person was an author of The Federalist, then he was a U.S. Founding Father. If James Madison was an author of The Federalist, then he was a U.S. Founding Father. Option C is correct for question 20.

Step2: Analyze question 21

The definition of a unicycle is a vehicle with a single wheel propelled by pedals. The reverse statement “A vehicle propelled by pedals is a unicycle” is incorrect because there are other pedal - propelled vehicles. It does not pass the reversibility test. Option B is correct for question 21.

Step3: Analyze question 22

A department store is a place to purchase clothing, but a place to purchase clothing is not necessarily a department store (it could be a boutique, etc.). The reverse statement does not pass the reversibility test. Option C is correct for question 22.

Step4: Prove question 23

1. Since $\overline{MO}$ bisects $\angle NPQ$, we have $m\angle NPO=m\angle OPQ$ (by the definition of angle - bisector).
2. Given that $\angle NPO$ and $\angle NPM$ are supplementary, so $m\angle NPO + m\angle NPM=180^{\circ}$. Also, $\angle OPQ$ and $\angle QPM$ are supplementary, so $m\angle OPQ + m\angle QPM = 180^{\circ}$.
3. From $m\angle NPO + m\angle NPM=180^{\circ}$ and $m\angle OPQ + m\angle QPM = 180^{\circ}$, and $m\angle NPO=m\angle OPQ$ (from step 1), we can substitute $m\angle NPO$ for $m\angle OPQ$ in the second equation.
4. We get $m\angle NPO + m\angle QPM = 180^{\circ}$. Since $m\angle NPO + m\angle NPM=180^{\circ}$, by the subtraction property of equality, $m\angle QPM=m\angle NPM$.

Answer:

1. C. If a person was an author of The Federalist, then he was a U.S. Founding Father. If James Madison was an author of The Federalist, then he was a U.S. Founding Father.
2. B. A vehicle propelled by pedals is a unicycle; does not pass
3. C. A place to purchase clothing is a department store; does not pass
4. Proven as above.