Question

given: ( \frac{ad}{dc}=\frac{be}{ec} )
prove: ( overline{ab}paralleloverline{de} )
complete the steps of the proof.

statements

1. ( \frac{ad}{dc}=\frac{be}{ec} )
2. ( \frac{ad}{dc}+1=\frac{be}{ec}+1 )
3. ( \frac{ad}{dc}+\frac{dc}{dc}=\frac{be}{ec}+\frac{ec}{ec} )
4. ( \frac{ad + dc}{dc}=\frac{be + ec}{ec} )
5. ( ac = ad + dc ); ( bc = be + ec )
6. ( \frac{ac}{bc}=\frac{dc}{ec} )
7. ( angle3=angle3 )
8. ( \triangle abcsim\triangle dec )
9. ( angle1=angle2 )

Explanation:

Step1: <Given>

$\frac{AD}{DC}=\frac{BE}{EC}$

Step2: <Add 1 to both sides>

$\frac{AD}{DC}+1=\frac{BE}{EC}+1$

Step3: <Simplify>

$\frac{AD + DC}{DC}=\frac{BE + EC}{EC}$

Step4: <Segment addition>

Since $AC = AD+DC$ and $BC=BE + EC$

Step5: <Substitute>

$\frac{AC}{BC}=\frac{EC}{DC}$

Step6: <Reflexive property>

$\angle3=\angle3$

Step7: <Similarity by SAS>

$\triangle ABC\sim\triangle DEC$

Step8: <Corresponding angles>

$\angle1=\angle2$

Step9: <Converse of corresponding angles>

$AB\parallel DE$

Answer:

$AB\parallel DE$ is proved.