Question

triangle congruence: sas
proving triangles congruent
given △dfe is isosceles with base fe, fb ≅ 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
segment df is congruent to segment de by the definition of isosceles triangle. since these
angles dce and dbf are congruent by the isosceles triangle theorem. therefore, triangles
dfb and dec are congruent by the sas congruence theorem.

Explanation:

Step1: Identify the property of isosceles triangle

Since \(\triangle DFE\) is isosceles with base \(FE\), by the definition of isosceles triangle, the legs \(DF\) and \(DE\) are congruent. So, \(DF\cong DE\).

Step2: Use the given congruent segments

We are given \(FB\cong EC\).

Step3: Use the reflexive property of angles

\(\angle DFB\) and \(\angle DEC\) are angles. \(\angle DFB\cong\angle DEC\) (If \(\triangle DFE\) is isosceles with base \(FE\), then \(\angle DFE=\angle DEF\), and using linear - pair or angle - subtraction concepts, we can get \(\angle DFB\cong\angle DEC\)).

Step4: Apply SAS (Side - Angle - Side) congruence criterion

In \(\triangle DFB\) and \(\triangle DEC\), we have \(DF\cong DE\) (from Step 1), \(\angle DFB\cong\angle DEC\) (from Step 3), and \(FB\cong EC\) (given). By the SAS (Side - Angle - Side) congruence criterion, \(\triangle DFB\cong\triangle DEC\).

Answer:

\(\triangle DFB\cong\triangle DEC\) by the SAS (Side - Angle - Side) congruence criterion.