Question

given: △dfe is isosceles with base fe; $overline{fb} cong overline{ec}$. prove: △dfb ≅ △dec
complete the missing parts of the paragraph proof.
we know that triangle dfe is isosceles with base 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.

Explanation:

Step1: Analyze FB and EC

Given \( \overline{FB} \cong \overline{EC} \) (given in the problem), so this is the reason for their congruence.

Step2: Analyze DF and DE

In isosceles triangle \( \triangle DFE \) with base \( FE \), the legs \( DF \) and \( DE \) are congruent by the definition of an isosceles triangle (two sides of an isosceles triangle are congruent). So \( DF \cong DE \).

Step3: Analyze angles

The base angles of an isosceles triangle are congruent, so \( \angle F \cong \angle E \).

Step4: Prove triangle congruence

Now, in \( \triangle DFB \) and \( \triangle DEC \), we have \( FB \cong EC \) (given), \( \angle F \cong \angle E \) (base angles of isosceles \( \triangle DFE \)), and \( DF \cong DE \) (legs of isosceles \( \triangle DFE \)). By SAS (Side - Angle - Side) congruence criterion, \( \triangle DFB \cong \triangle DEC \).

Filling in the blanks:

• First blank (why \( FB \cong EC \)): Given (since it's stated \( \overline{FB} \cong \overline{EC} \) in the problem).
• Second blank (what \( DF \cong \) to): \( DE \) (by definition of isosceles triangle).
• Third blank (which angles are congruent): \( \angle F \) and \( \angle E \) (base angles of isosceles triangle).
• Fourth blank (which triangles are congruent): \( \triangle DFB \) and \( \triangle DEC \) (the two triangles we are proving congruent).

Answer:

• First blank: Given
• Second blank: \( DE \)
• Third blank: \( \angle F \) and \( \angle E \)
• Fourth blank: \( \triangle DFB \) and \( \triangle DEC \)