Question

given: δdfe is isosceles with base fe, fb ≅ ec. prove: δdfb ≅ δdec. 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. first, since blank, segment df is congruent to segment blank by the blank. definition of isosceles triangle: since these segments are congruent, the base angles, angles blank, are congruent by the isosceles triangle theorem. therefore, triangles blank are congruent by sas.

Explanation:

Step1: Identify Given Information

We know $\triangle DFE$ is isosceles with base $FE$, so by the definition of an isosceles triangle, the legs opposite the base are equal. Thus, $DF = DE$. Also, given $FB\cong EC$.

Step2: Analyze Congruent Sides and Angles

• Side 1: We are given $FB\cong EC$ (given).
• Side 2: Since $\triangle DFE$ is isosceles with base $FE$, $DF = DE$ (legs of isosceles triangle).
• Angle: The base angles of an isosceles triangle are equal, so $\angle F\cong\angle E$ (base angles of isosceles $\triangle DFE$).

Step3: Apply SAS Congruence Criterion

To prove $\triangle DFB\cong\triangle DEC$, we use the Side - Angle - Side (SAS) congruence postulate. We have:

• $DF = DE$ (from isosceles triangle property)
• $\angle F=\angle E$ (base angles of isosceles triangle)
• $FB = EC$ (given)

So, by SAS, $\triangle DFB\cong\triangle DEC$.

Now, filling in the proof:

1. We know that triangle $DFE$ is isosceles with base $FE$ and that segment $FB$ is congruent to segment $EC$ (Given).
2. Segment $DF$ is congruent to segment $DE$ (by the definition of an isosceles triangle: legs of an isosceles triangle are congruent).
3. Definition of base - angle theorem: Since these segments (the legs) are congruent, the base angles, angles $\angle F$ and $\angle E$, are congruent (by the base - angle theorem of isosceles triangles).
4. Triangle Reason: Therefore, triangles $\triangle DFB$ and $\triangle DEC$ are congruent by SAS (Side - Angle - Side) congruence postulate.

Answer:

1. Given
2. $DE$; Definition of isosceles triangle
3. $\angle F$; $\angle E$; Base - angle theorem
4. $\triangle DFB$; $\triangle DEC$; SAS