Exercise 20 Page 574

We will write a two-column proof of the Segments of Secants Theorem (Theorem 10.19). Let's start by reviewing what it is.

To prove this theorem we will use the following diagram.

Note that two-column proof lists each statement on the left and the justification on the right. Each statement must follow logically from the steps before it. In a two-column proof we always start by stating the given information. Statement1)& EB and ED are secant & segments that share the same & endpoint outside a circle. Reason1)& GivenTo continue our proof we will write a similarity relation between △ EBC and △ EDA. Therefore, to have △ EBC and △ EDA we need to draw BC and AD.

We can draw these segments by the definition of a segment. Statement2)& Draw BC and AD. Reason2)& Definition of a Segment Now that we have two triangles we can compare them. Notice that both triangles share vertex E.

By the Reflexive Property of Equality, ∠ E is congruent to itself. Statement3)& ∠ E ≅ ∠ E. Reason3)& Reflexive Property & of Equality Now, let's look at ∠ EBC and ∠ EDA. Both of them intercept the same arc, AC.

By the Inscribed Angles of a Circle Theorem (Theorem 10.11), we know that if two inscribed angles of a circle intercept the same arc, then the angles are congruent. Therefore, ∠ EBC and ∠ EDA are congruent. Statement4)& ∠ EBC ≅ ∠ EDA. Reason4)& Inscribed Angles of & a Circle Theorem With this step, the two angles of △ EBC are congruent to the corresponding two angles of △ EDA. From here, by the Angle-Angle (AA) Similarity Theorem (Theorem 8.3) we can conclude that the triangles are similar. Statement5)& △ EBC ~ △ EDA Reason5)& AA Similarity Theorem Since the corresponding side lengths of congruent triangles are proportional, we can write the following proportion. Statement6)& EA/EC=ED/EB Reason6)& Corresponding side lengths & of congruent triangles & are proportional. Finally, using the Cross Product Property we will complete the proof. Statement7)& EA * EB= EC * ED Reason7)& Cross Product Property Let's summarize the above process in a two-column table.