Question

question
given: abcd is a rhombus and ∠bae ≅ ∠bcf.
prove: ed ≅ fd.
step
statement
reason
1
abcd is a rhombus

2
∠bae ≅ ∠bcf
given
3
∠dab ≅ ∠dcb
select a reason...

Explanation:

Step1: Properties of a rhombus

In a rhombus \(ABCD\), \(AB = BC=CD = DA\) and \(\angle DAB=\angle DCB\) (opposite angles of a rhombus are equal).

Step2: Angle - subtraction

Since \(\angle BAE\cong\angle BCF\), we can use the property that \(\angle DAB-\angle BAE=\angle DCB - \angle BCF\). So, \(\angle DAE=\angle DCF\)

Step3: Side - Angle - Side (SAS) congruence

In \(\triangle ADE\) and \(\triangle CDF\):

• \(AD = CD\) (sides of a rhombus)
• \(\angle DAE=\angle DCF\) (from step 2)
• \(\angle ADE=\angle CDF\) (vertically - opposite angles)

By the SAS congruence criterion, \(\triangle ADE\cong\triangle CDF\)

Step4: Corresponding parts of congruent triangles

If \(\triangle ADE\cong\triangle CDF\), then \(ED = FD\) (corresponding parts of congruent triangles are equal)

Answer:

\(\overline{ED}\cong\overline{FD}\)