Question

given: δdfe is isosceles with base \\(\overline{fe}\\); \\(\overline{fb} \cong \overline{ec}\\). prove: \\(\delta dfb \cong \delta dec\\) diagram: triangle dfe with f, c, b, e on the base (fe), d at the vertex. complete the missing parts of the paragraph proof. we know that triangle dfe is isosceles with base \\(\overline{fe}\\) and that segment fb is congruent to segment ec because dropdown. segment df is congruent to segment dropdown by the definition of isosceles triangle. since these segments are congruent, the base angles, angles dropdown, are congruent by the isosceles triangle theorem. therefore, triangles dropdown are congruent by sas. options for last dropdown: dfb and dec, ecd and dbc, fdc and edb

Explanation:

Step 1: Analyze the first blank (reason for \( \overline{FB} \cong \overline{EC} \))

The problem states \( \overline{FB} \cong \overline{EC} \) as given, so the first blank is "it is given".

Step 2: Analyze the second blank (congruent to \( \overline{DF} \))

In isosceles \( \triangle DFE \) with base \( \overline{FE} \), the legs \( \overline{DF} \) and \( \overline{DE} \) are congruent (by definition of isosceles triangle). So the second blank is \( \overline{DE} \).

Step 3: Analyze the third blank (congruent base angles)

The base angles of \( \triangle DFE \) are \( \angle F \) and \( \angle E \) (since base is \( \overline{FE} \)). By the isosceles triangle theorem, these angles are congruent. So the third blank is \( \angle F \) and \( \angle E \).

Step 4: Analyze the fourth blank (triangles congruent by SAS)

We have:

• \( \overline{DF} \cong \overline{DE} \) (from Step 2),
• \( \angle F \cong \angle E \) (from Step 3),
• \( \overline{FB} \cong \overline{EC} \) (given).

By SAS, \( \triangle DFB \cong \triangle DEC \). So the fourth blank is "DFB and DEC".

Final Answers for Each Blank:

1. First blank: it is given
2. Second blank: \( \overline{DE} \)
3. Third blank: \( \angle F \) and \( \angle E \)
4. Fourth blank: DFB and DEC

(Note: For the dropdowns, select the corresponding options as above.)

Answer:

Step 1: Analyze the first blank (reason for \( \overline{FB} \cong \overline{EC} \))

The problem states \( \overline{FB} \cong \overline{EC} \) as given, so the first blank is "it is given".

Step 2: Analyze the second blank (congruent to \( \overline{DF} \))

In isosceles \( \triangle DFE \) with base \( \overline{FE} \), the legs \( \overline{DF} \) and \( \overline{DE} \) are congruent (by definition of isosceles triangle). So the second blank is \( \overline{DE} \).

Step 3: Analyze the third blank (congruent base angles)

The base angles of \( \triangle DFE \) are \( \angle F \) and \( \angle E \) (since base is \( \overline{FE} \)). By the isosceles triangle theorem, these angles are congruent. So the third blank is \( \angle F \) and \( \angle E \).

Step 4: Analyze the fourth blank (triangles congruent by SAS)

We have:

• \( \overline{DF} \cong \overline{DE} \) (from Step 2),
• \( \angle F \cong \angle E \) (from Step 3),
• \( \overline{FB} \cong \overline{EC} \) (given).

By SAS, \( \triangle DFB \cong \triangle DEC \). So the fourth blank is "DFB and DEC".

Final Answers for Each Blank:

1. First blank: it is given
2. Second blank: \( \overline{DE} \)
3. Third blank: \( \angle F \) and \( \angle E \)
4. Fourth blank: DFB and DEC

(Note: For the dropdowns, select the corresponding options as above.)