Question

given: \\( \frac { a d } { d c } = \frac { b e } { e c } \\) prove: \\( \overline { a b } \parallel \overline { d e } \\) complete the steps of the proof. 1. \\( \frac { a d } { d c } = \frac { b e } { e c } \\) 1. given 2. \\( \frac { a d } { d c } + 1 = \frac { b e } { e c } + 1 \\) 2. addition property 3. \\( \frac { a d } { d c } + \frac { d c } { d c } = \frac { b e } { e c } + \frac { e c } { e c } \\) 3. property of proportion 4. \\( \frac { a d + d c } { d c } = \frac { b e + e c } { e c } \\) 4. addition of fractions 5. \\( a c = a d + d c \\); \\( b c = b e + e c \\) 5. segment addition postulate 6. \\( \frac { a c } { d c } = \frac { b c } { e c } \\) 6. substitution property 7. \\( \angle 3 = \angle 3 \\) 7. reflexive property 8. \\( \triangle a b c \sim \triangle d e c \\) 8. 9. \\( \angle 1 = \angle 2 \\) 9. definition of congruent angles definition of correspond - ing angles definition of similar triangles

Explanation:

Step1: Analyze the given proportion

Given \(\frac{AD}{DC}=\frac{BE}{EC}\).

Step2: Use addition property of equality

Add \(1\) to both sides: \(\frac{AD}{DC}+1=\frac{BE}{EC}+1\).

Step3: Simplify the left - hand side

\(\frac{AD + DC}{DC}=\frac{BE+EC}{EC}\) (using the property \(\frac{a}{b}+1=\frac{a + b}{b}\)).

Step4: Use segment addition postulate

Since \(AC=AD + DC\) and \(BC=BE + EC\) (segment addition postulate).

Step5: Substitute

We get \(\frac{AC}{DC}=\frac{BC}{EC}\) (substitution property).

Step6: Check for similar triangles

\(\angle3=\angle3\) (reflexive property).

Step7: Apply SAS similarity theorem

In \(\triangle ABC\) and \(\triangle DEC\), we have \(\frac{AC}{DC}=\frac{BC}{EC}\) and \(\angle3\) is common. So, \(\triangle ABC\sim\triangle DEC\) (SAS similarity theorem).

Step8: Use definition of similar triangles

If \(\triangle ABC\sim\triangle DEC\), then \(\angle1=\angle2\) (corresponding angles of similar triangles).

Step9: Prove parallel lines

Since \(\angle1\) and \(\angle2\) are alternate interior angles and \(\angle1=\angle2\), by the converse of the alternate interior angles theorem, \(AB\parallel DE\).

Answer:

1. given
2. addition property
3. property of proportion
4. addition of fractions
5. segment addition
6. substitution property
7. reflexive property
8. SAS similarity theorem
9. definition of similar triangles