Question

statements reasons

1. \overleftrightarrow{ab} \parallel \overleftrightarrow{cg} given
2. construct a perpendicular line \overleftrightarrow{fc} so we can see triangles
3. in $\triangle dbf$,

$m\angle dbf + m\angle dfb + 90^\circ = 180^\circ$ triangle sum theorem

1. $m\angle bdf = 90^\circ - m\angle dfb$ subtraction property of equality
2. in $\triangle egf$,

$m\angle gef + m\angle efg + 90^\circ = 180^\circ$ triangle sum theorem

1. $m\angle gef = 90^\circ - m\angle efc$

subtraction property of equality

1. $m\angle dfb = m\angle efc$
2. $m\angle bdf = m\angle gef$ transitive property of equality
3. $\angle bdf \cong \angle gef$

try again.

Explanation:

To determine the correct reason for step 8 (\(m\angle BDF = m\angle GEF\)), we analyze the previous steps:

• Step 4: \(m\angle BDF = 90^\circ - m\angle DFB\)
• Step 6: \(m\angle GEF = 90^\circ - m\angle EFG\)
• Step 7: \(m\angle DFB = m\angle EFG\) (vertical angles or equal angles, though here it’s given or derived)

The Transitive Property of Equality states that if \(a = b\) and \(b = c\), then \(a = c\). Substituting \(m\angle DFB\) (from step 7) into the expressions for \(m\angle BDF\) and \(m\angle GEF\) (from steps 4 and 6) gives \(m\angle BDF = m\angle GEF\).

Answer:

Transitive Property of Equality