Exercise 39 Page 669

Let's begin by marking the parallel segments in the given diagram.

By the Triangle Proportionality Theorem, we obtain the relation below. AD/DB = EC/BC By applying the Alternate Interior Angles Theorem and the Corresponding Angles Postulate, we have that ∠ 1 ≅ ∠ 3 and ∠ 2 ≅ ∠ E respectively.

Additionally, ∠ 1 ≅ ∠ 2 by definition of angle bisector. Consequently, we have that ∠ 3 ≅ ∠ E. Thus, by the Converse Isosceles Triangle Theorem, we get that EC ≅ AC.

By definition of congruent segments, we have that EC = AC. Finally, let's substitute it into the initial equation. AD/DB = AC/BC

Two-Column Proof Table

Let's summarize the proof we just did in the following table.

Alternative Solution

Alternative Solution

We begin by highlighting the congruent angles in the given diagram.

Since CD is the bisector of ∠ ACB, we can apply the Triangle Angle Bisector (Theorem 7.11) and get the required relation directly. AD/DB = AC/BC