Question

in the figure shown, $overleftrightarrow{cf}$ intersects $overleftrightarrow{ad}$ and $overleftrightarrow{eh}$ at points $b$ and $f$, respectively
given: $m\angle cbd = m\angle bfe$
prove: $m\angle bfe + m\angle dbf = 180^\circ$
statement: $m\angle cbd = m\angle bfe$ reason: choose
statement: $m\angle cbd + m\angle dbf = 180^\circ$ reason: choose
statement: $m\angle bfe + m\angle dbf = 180^\circ$ reason: choose

Explanation:

Step1: Identify Given and Goal

Given \( m\angle CBD = m\angle BFE \), need to prove \( m\angle BFE + m\angle DBF = 180^\circ \).

Step2: Analyze Linear Pair

\( \angle CBD \) and \( \angle DBF \) form a linear pair, so \( m\angle CBD + m\angle DBF = 180^\circ \) (linear pair postulate: adjacent supplementary angles sum to \( 180^\circ \)).

Step3: Substitute Given Equality

Since \( m\angle CBD = m\angle BFE \) (given), substitute \( \angle CBD \) with \( \angle BFE \) in the linear pair equation: \( m\angle BFE + m\angle DBF = 180^\circ \).

Answer:

To prove \( m\angle BFE + m\angle DBF = 180^\circ \):

1. Statement: \( m\angle CBD + m\angle DBF = 180^\circ \)

Reason: Linear Pair Postulate (adjacent angles forming a straight line sum to \( 180^\circ \)).

1. Statement: \( m\angle CBD = m\angle BFE \)

Reason: Given.

1. Statement: \( m\angle BFE + m\angle DBF = 180^\circ \)

Reason: Substitution Property (replace \( \angle CBD \) with \( \angle BFE \) in the first equation).

Final conclusion: \( \boldsymbol{m\angle BFE + m\angle DBF = 180^\circ} \) is proven.