Question

prove fe + de > df
statements

1. δdef
2. draw auxiliary lines eg and dg so that eg = de
3. fe + ge = fg
4. ∠1 ≅ ∠2
5. m∠1 = m∠2
6. m∠gdf = m∠2 + m∠3
7. m∠gdf > m∠2
8. m∠gdf > m∠1
9. fg > df
10. fe + ge > df
11. fe + de > df

reasons

1. given
2. through any two points there is exactly one line.
3. addition postulate
4. isosceles δ theorem

5.

1. postulate
2. comparison property of ≠

8.

1. in a δ, the longer side is opposite the larger ∠
2. substitution (from line 3)
3. (from line 2)

a. property
b. substitution
c. substitution property
d. segmente. angle additionf. def. ≅ anglesg. def. isosceles δh. transitive property of substitution

Explanation:

Step 1: Analyze Line 3

The statement is \( FE + GE = FG \), which is about segment addition. The Segment Addition Postulate states that if a point \( G \) is between \( F \) and \( G \) (wait, correction: if a point \( E \) is between \( F \) and \( G \)? Wait, no, the formula \( FE + GE = FG \) implies that \( E \) is a point such that \( F - E - G \) (collinear), so the reason is the Segment Addition Postulate. So for line 3, the reason is "d. Segment (Addition Postulate)".

Step 2: Analyze Line 5

\( \angle 1 \cong \angle 2 \) implies \( m\angle 1 = m\angle 2 \) by the Definition of Congruent Angles (Def. \( \cong \) angles), so reason is "f. Def. \( \cong \) angles".

Step 3: Analyze Line 6

\( m\angle GDF = m\angle 2 + m\angle 3 \) is the Angle Addition Postulate, which states that if a ray \( DG \) is inside \( \angle GDF \) (wait, correction: if a ray \( DE \) splits \( \angle GDF \) into \( \angle 2 \) and \( \angle 3 \)), so the reason is "e. Angle addition (Postulate)".

Step 4: Analyze Line 8

We have \( m\angle GDF > m\angle 2 \) (line 7) and \( m\angle 1 = m\angle 2 \) (line 5), so by the Transitive Property of Inequality (or Substitution), \( m\angle GDF > m\angle 1 \). The reason is "h. Transitive Property of Substitution" (since we substitute \( m\angle 2 \) with \( m\angle 1 \) in the inequality).

Step 5: Analyze Line 10

We have \( FG > DF \) (line 9) and \( FE + GE = FG \) (line 3), so substituting \( FG \) with \( FE + GE \) gives \( FE + GE > DF \). The reason is substitution, and the property is using the equation from line 3, so it's "c. Substitution Property" (wait, line 10's reason is "Substitution (from line 3)", so the option is "c. Substitution Property"? Wait, the options for line 10: the dropdown is for the property. Wait, line 10: \( FE + GE > DF \) because \( FG > DF \) and \( FG = FE + GE \), so substituting \( FG \) with \( FE + GE \) is the Substitution Property. So reason is "c. Substitution Property".

Step 6: Analyze Line 11

We have \( GE = DE \) (line 2), so substituting \( GE \) with \( DE \) in \( FE + GE > DF \) gives \( FE + DE > DF \). The reason is Substitution (from line 2), so the option is "b. Substitution" (since we substitute \( GE \) with \( DE \)).

Wait, let's recheck each line:

Line 3: \( FE + GE = FG \) → Segment Addition Postulate → option d.

Line 5: \( m\angle 1 = m\angle 2 \) → Def. of congruent angles → option f.

Line 6: \( m\angle GDF = m\angle 2 + m\angle 3 \) → Angle Addition Postulate → option e.

Line 8: \( m\angle GDF > m\angle 1 \) → since \( m\angle GDF > m\angle 2 \) and \( m\angle 1 = m\angle 2 \), transitive property → option h.

Line 10: Substitution from line 3 → using the equation \( FE + GE = FG \) to substitute into \( FG > DF \), so Substitution Property → option c.

Line 11: Substitution from line 2 (since \( GE = DE \)) → substitute \( GE \) with \( DE \), so option b.

But the question is probably asking for each blank:

Line 3 Reason: d (Segment Addition Postulate)

Line 5 Reason: f (Def. \( \cong \) angles)

Line 6 Reason: e (Angle Addition Postulate)

Line 8 Reason: h (Transitive Property of Substitution)

Line 10 Reason: c (Substitution Property)

Line 11 Reason: b (Substitution)

But let's focus on the blanks:

3. The reason for \( FE + GE = FG \) is the Segment Addition Postulate, so option d.

5. \( \angle 1 \cong \angle 2 \) implies \( m\angle 1 = m\angle 2 \) by Def. of Congruent Angles (f).

6. \( m\angle GDF = m\angle 2 + m\angle 3 \) is Angle Addition Postulate (e).

8. \( m\angle GDF > m\angle 1 \) uses transitive property (h) because \( m\angle GDF > m\angle 2 \) and \( m\angle 1 = m\angle 2 \).

10. Substitution from line 3 uses Substitution Property (c).

11. Substitution from line 2 (GE = DE) uses Substitution (b).

But maybe the question is for a specific line? Wait, the original problem has blanks for lines 3,5,6,8,10,11. Let's assume we need to fill each:

Line 3: Reason is "d. Segment" (Segment Addition Postulate)

Line 5: Reason is "f. Def. \( \cong \) angles"

Line 6: Reason is "e. Angle addition"

Line 8: Reason is "h. Transitive Property of Substitution"

Line 10: Reason is "c. Substitution Property"

Line 11: Reason is "b. Substitution"

Answer:

For each line:

1. d. Segment
2. f. Def. \( \cong \) angles
3. e. Angle addition
4. h. Transitive Property of Substitution
5. c. Substitution Property
6. b. Substitution

(If focusing on a specific line, e.g., line 3: d. Segment, line 5: f. Def. \( \cong \) angles, etc. Based on the options, the answers are as above.)