segment addition ws page 1 2 .pdf - Geometry 1.2 Segments ... - Free Printable

Problem Analysis:

The problem involves a geometric figure with two parallel lines, \( AB \) and \( CD \), intersected by a transversal. The goal is to find the length of segment \( EF \) using the given information about other segments.

#### Given:
1. \( AB \parallel CD \)
2. \( AE = 6 \) cm
3. \( EB = 9 \) cm
4. \( DF = 8 \) cm

#### To Find:
The length of segment \( EF \).

---

Step-by-Step Solution:

#### Step 1: Understand the Geometry
Since \( AB \parallel CD \), the transversal creates corresponding angles and similar triangles. Specifically, the triangles formed by the transversal and the parallel lines are similar due to the properties of parallel lines and transversals.

The key similarity here is between \( \triangle AEB \) and \( \triangle DEF \). This similarity arises because:
- \( \angle AEB = \angle DEF \) (corresponding angles)
- \( \angle EAB = \angle EDF \) (alternate interior angles)

Thus, \( \triangle AEB \sim \triangle DEF \) by the AA (Angle-Angle) similarity criterion.

#### Step 2: Use the Similarity Ratio
Since the triangles are similar, the ratios of corresponding sides are equal. Therefore:
\[
\frac{AE}{DE} = \frac{EB}{EF}
\]

#### Step 3: Express Known Quantities
From the problem:
- \( AE = 6 \) cm
- \( EB = 9 \) cm
- \( DF = 8 \) cm

We need to find \( EF \). Let \( DE = x \). Then, the total length of \( AD \) can be expressed as:
\[
AD = AE + ED = 6 + x
\]

#### Step 4: Set Up the Proportion
Using the similarity ratio:
\[
\frac{AE}{DE} = \frac{EB}{EF}
\]
Substitute the known values:
\[
\frac{6}{x} = \frac{9}{EF}
\]

#### Step 5: Solve for \( EF \)
Cross-multiply to solve for \( EF \):
\[
6 \cdot EF = 9 \cdot x
\]
\[
EF = \frac{9x}{6} = \frac{3x}{2}
\]

#### Step 6: Use Additional Information
We also know that \( DF = 8 \) cm. Since \( D \) is a point on the line \( CD \), and \( F \) is a point on the transversal, the segment \( DF \) is part of the larger segment \( CD \). However, the problem does not provide the full length of \( CD \), so we focus on the relationship between the segments.

#### Step 7: Revisit the Proportion
From the similarity, we have:
\[
\frac{AE}{DE} = \frac{EB}{EF}
\]
Substitute \( AE = 6 \), \( EB = 9 \), and \( EF = \frac{3x}{2} \):
\[
\frac{6}{x} = \frac{9}{\frac{3x}{2}}
\]
Simplify the right-hand side:
\[
\frac{9}{\frac{3x}{2}} = \frac{9 \cdot 2}{3x} = \frac{18}{3x} = \frac{6}{x}
\]

This confirms the proportion is consistent. Now, we need to find \( x \) using the given \( DF = 8 \) cm. Since \( DF \) is part of the larger segment and the triangles are similar, we can use the ratio directly:
\[
\frac{AE}{DE} = \frac{EB}{EF}
\]
\[
\frac{6}{x} = \frac{9}{EF}
\]
Given \( DF = 8 \) cm, we can infer:
\[
EF = \frac{3 \cdot 8}{2} = 12 \text{ cm}
\]

#### Final Answer:
\[
\boxed{12}
\]