Question

given: bcde is a parallelogram, ∠bae ≅ ∠cde, overline{ae} ≅ overline{cd} prove: △ abe is equilateral drag each necessary statement to the table to complete the proof. statements reasons bcde is a parallelogram, ∠bae ≅ ∠cde, overline{ae} ≅ overline{cd} given opposite sides of a parallelogram are congruent transitive property corresponding angles theorem transitive property converse of isosceles triangle theorem transitive property △ abe is equilateral definition of equilateral overline{ae} ≅ overline{be} overline{bc} ≅ overline{ed} overline{be} ≅ overline{ab} overline{cd} ≅ overline{be} overline{ae} ≅ overline{be} ≅ overline{ab} ∠bae ≅ ∠bea ∠bcd ≅ ∠deb ∠cde ≅ ∠bea ∠cde ≅ ∠ebc overline{cd} ≅ overline{be} ≅ overline{ab} question #31

Explanation:

Step1: Match parallelogram side congruence

$\overline{BC} \cong \overline{ED}$, $\overline{CD} \cong \overline{BE}$

Step2: Transitive property for $\overline{AE} \cong \overline{BE}$

Given $\overline{AE} \cong \overline{CD}$ and $\overline{CD} \cong \overline{BE}$, so $\overline{AE} \cong \overline{BE}$

Step3: Corresponding angles from parallelogram

Since $BC \parallel ED$, $\angle CDE \cong \angle BEA$

Step4: Transitive property for $\angle BAE \cong \angle BEA$

Given $\angle BAE \cong \angle CDE$ and $\angle CDE \cong \angle BEA$, so $\angle BAE \cong \angle BEA$

Step5: Converse of isosceles triangle theorem

From $\angle BAE \cong \angle BEA$, $\overline{BE} \cong \overline{AB}$

Step6: Transitive property for all sides congruent

From $\overline{AE} \cong \overline{BE}$ and $\overline{BE} \cong \overline{AB}$, so $\overline{AE} \cong \overline{BE} \cong \overline{AB}$

Answer:

The completed proof table is:

Statements	Reasons
$\overline{CD} \cong \overline{BE}$	Opposite sides of a parallelogram are congruent
$\overline{AE} \cong \overline{BE}$	Transitive Property
$\angle CDE \cong \angle BEA$	Corresponding Angles Theorem
$\angle BAE \cong \angle BEA$	Transitive Property
$\overline{BE} \cong \overline{AB}$	Converse of Isosceles Triangle Theorem
$\overline{AE} \cong \overline{BE} \cong \overline{AB}$	Transitive Property
$\triangle ABE$ is equilateral	Definition of Equilateral