Question

give a reason for each statement.
f r a n
given: $overline{fr} \cong \overline{an}$
prove: $overline{fa} \cong \overline{rn}$
\

$$\begin{tabular}{|l|l|} \\hline statements & reasons \\\\ \\hline 1. $\\overline{fr} \\cong \\overline{an}$ & 1. \\\\ \\hline 2. $fr = an$ & 2. \\\\ \\hline 3. $ra = ra$ & 3. \\\\ \\hline 4. $fr + ra = an + ra$ & 4. \\\\ \\hline 5. $fr + ra = fa$ & 5. \\\\ \\hline 6. $an + ra = rn$ & 6. \\\\ \\hline 7. $fa = rn$ & 7. \\\\ \\hline 8. $\\overline{fa} \\cong \\overline{rn}$ & 8. \\\\ \\hline \\end{tabular}$$

given \quad def. of congruent \quad seg. add. post. \quad angle add. post. \quad add. prop \quad subt. prop \quad reflex. prop \quad symm. prop \quad trans. prop

1 \quad (radio button selected for \given\)
2 \quad (radio button)
3 \quad (radio button)
4 \quad (radio button)

Explanation:

Step-by-Step Explanation for Each Statement:

Statement 1: $\overline{FR} \cong \overline{AN}$

Step 1: Identify the Reason

This is given in the problem statement.
Reason: given

Statement 2: $FR = AN$

Step 2: Recall Congruent Segments

If two segments are congruent, their lengths are equal (definition of congruent segments).
Reason: def. of congruent

Statement 3: $RA = RA$

Step 3: Reflexive Property

A segment is congruent (and equal) to itself (reflexive property of equality).
Reason: reflex. prop

Statement 4: $FR + RA = AN + RA$

Step 4: Addition Property of Equality

If $a = b$, then $a + c = b + c$ (addition property of equality, applied to $FR = AN$ and $c = RA$).
Reason: add. prop

Statement 5: $FR + RA = FA$

Step 5: Segment Addition Postulate

If $R$ is between $F$ and $A$, then $FR + RA = FA$ (segment addition postulate).
Reason: seg. add. post.

Statement 6: $AN + RA = RN$

Step 6: Segment Addition Postulate

If $A$ is between $R$ and $N$, then $AN + RA = RN$ (segment addition postulate).
Reason: seg. add. post.

Statement 7: $FA = RN$

Step 7: Substitution (Transitive/Equality)

From Steps 5 and 6, $FR + RA = FA$ and $AN + RA = RN$, and from Step 4, $FR + RA = AN + RA$. Thus, $FA = RN$ (substitution or transitive property of equality).
Reason: trans. prop (or substitution, but transitive fits as $FA = FR + RA$ and $RN = AN + RA$, and $FR + RA = AN + RA$ implies $FA = RN$)

Statement 8: $\overline{FA} \cong \overline{RN}$

Step 8: Definition of Congruent Segments

If two segments have equal length, they are congruent (definition of congruent segments, reversing Step 2).
Reason: def. of congruent

Final Answers (Reasons for Each Statement):

1. given
2. def. of congruent
3. reflex. prop
4. add. prop
5. seg. add. post.
6. seg. add. post.
7. trans. prop (or substitution)
8. def. of congruent

(Note: For the multiple-choice bubbles, the selected options would correspond to these reasons.)

Answer:

Step-by-Step Explanation for Each Statement:

Statement 1: $\overline{FR} \cong \overline{AN}$

Step 1: Identify the Reason

This is given in the problem statement.
Reason: given

Statement 2: $FR = AN$

Step 2: Recall Congruent Segments

If two segments are congruent, their lengths are equal (definition of congruent segments).
Reason: def. of congruent

Statement 3: $RA = RA$

Step 3: Reflexive Property

A segment is congruent (and equal) to itself (reflexive property of equality).
Reason: reflex. prop

Statement 4: $FR + RA = AN + RA$

Step 4: Addition Property of Equality

If $a = b$, then $a + c = b + c$ (addition property of equality, applied to $FR = AN$ and $c = RA$).
Reason: add. prop

Statement 5: $FR + RA = FA$

Step 5: Segment Addition Postulate

If $R$ is between $F$ and $A$, then $FR + RA = FA$ (segment addition postulate).
Reason: seg. add. post.

Statement 6: $AN + RA = RN$

Step 6: Segment Addition Postulate

If $A$ is between $R$ and $N$, then $AN + RA = RN$ (segment addition postulate).
Reason: seg. add. post.

Statement 7: $FA = RN$

Step 7: Substitution (Transitive/Equality)

From Steps 5 and 6, $FR + RA = FA$ and $AN + RA = RN$, and from Step 4, $FR + RA = AN + RA$. Thus, $FA = RN$ (substitution or transitive property of equality).
Reason: trans. prop (or substitution, but transitive fits as $FA = FR + RA$ and $RN = AN + RA$, and $FR + RA = AN + RA$ implies $FA = RN$)

Statement 8: $\overline{FA} \cong \overline{RN}$

Step 8: Definition of Congruent Segments

If two segments have equal length, they are congruent (definition of congruent segments, reversing Step 2).
Reason: def. of congruent

Final Answers (Reasons for Each Statement):

1. given
2. def. of congruent
3. reflex. prop
4. add. prop
5. seg. add. post.
6. seg. add. post.
7. trans. prop (or substitution)
8. def. of congruent

(Note: For the multiple-choice bubbles, the selected options would correspond to these reasons.)