  ### Home > GC > Chapter 8 > Lesson 8.1.2 > Problem8-20

8-20.

Suzette started to set up a proof to show that if $\overline { B C } / / \overline { E F }$, $\overline { A B } / / \overline { D E }$ and $AF = DC$, then $\overline { B C } \cong \overline { E F }$. Examine her work below. Then complete her missing statements and reasons. Statements Reasons $\overline { B C } / / \overline { E F }$, $\overline { A B } / / \overline { D E }$ and $AF = DC$ $m\angle BCF = m\angle EFC$ and $m\angle EDF = m\angle CAB$ Reflexive Property $AF + FC = FC + DC$ Additive Property of Equality (adding the same amount to both sides of an equation keeps the equation true) $AC = DF$ Segment addition $\Delta ABC \cong \Delta DEF$ $\cong \Delta \text{s} \rightarrow \text{ parts}$

 Statements Reasons $\overline{BC} // \overline{EF}, \overline{AB}// \overline{DE}$, and $AF = DC$ Given $m\angle BCG = m\angle EFC$ and $m\angle EDF = m\angle CAB$ If two lines cut by a transversal are parallel, then alternate interior angles are equal. $\textbf { }\boldsymbol{FC} \mathbf {\;=} \boldsymbol{\:FC} \textbf{ }$ Reflexive Property $AF + FC = FC + DC$ Additive Property of Equality (adding the same amount to both sides of an equation keeps the equation true) $AC = DF$ Segment addition $\Delta ABC \cong \Delta DEF$ ASA $\boldsymbol {\cong}$ $\boldsymbol{\overline{BC}} \boldsymbol{\cong} \boldsymbol{\overline{EF}}$ $\cong \Delta\text{s} \rightarrow \; \cong$ parts