Question

in the following triangle, \\(\frac{ec}{ae} = \frac{db}{ad}\\).

below is the proof that \\(\overline{ed} \parallel \overline{cb}\\). the proof is divided into two parts, where the title of each part indicates its main purpose.

complete part b of the proof.

part a: prove \\(\frac{ac}{ae} = \frac{ab}{ad}\\)

show the steps.

part b: prove \\(\overline{ed} \parallel \overline{cb}\\)

statement	reason
9 \\(\triangle aed \sim \triangle\\) abc	side - side - side similarity (part a, 8)
10 \\(\angle 1 \cong \angle 2\\)	measures of corresponding angles of similar triangles are equal. (9)
11 \\(\overline{ed} \parallel \overline{cb}\\)	if a transversal crosses two lines and corresponding angles are congruent, then the lines are parallel. (10)

Explanation:

Step1: Confirm reflexive angle

$\angle A \cong \angle A$

Step2: Prove triangle similarity

$\triangle AED \sim \triangle ABC$ (by side-side-side similarity, using $\frac{AC}{AE}=\frac{AB}{AD}$ from Part A and $\angle A \cong \angle A$)

Step3: State congruent corresponding angles

$\angle 1 \cong \angle 2$ (corresponding angles of similar triangles are congruent)

Step4: Conclude parallel lines

$\overline{ED} \parallel \overline{CB}$ (if corresponding angles from a transversal are congruent, lines are parallel)

Answer:

The completed Part B proof is:

1. $\boldsymbol{\angle A \cong \angle A}$ | Reflexive property
2. $\boldsymbol{\triangle AED \sim \triangle ABC}$ | Side-side-side similarity (Part A, 8)
3. $\boldsymbol{\angle 1 \cong \angle 2}$ | Measures of corresponding angles of similar triangles are equal. (9)
4. $\boldsymbol{\overline{ED} \parallel \overline{CB}}$ | If a transversal crosses two lines and corresponding angles are congruent, then the lines are parallel. (10)