Exercise 121 Page 203

a If the given triangles are similar they will have three pairs of congruent corresponding angles. However, we only have to find two congruent pairs, as the third pair will then inevitably be congruent as well.

We start by recognizing that ∠ ACB and ∠ ECD are vertical angles, which means they are congruent by the Vertical Angles Theorem.

Next, if we view AE as a transversal to AB and DE, we can identify a pair of alternate interior angles. Since AB∥ DE, we know that ∠ A ≅ ∠ E by the Alternate Interior Angles Theorem.

With this information we know that △ ACB~ △ ECD by the AA~ condition. Let's show this as a flowchart.

b Let's add the given information in the exercise to the diagram.

In Part A we have shown that we have the following similarity statement. Note that the order of letters in the similarity statement allows us to tell which vertices are corresponding to each other.

△ ACB~ △ ECD Therefore, sides AC, CB, and AB correspond to EC, CD, and ED, respectively. Let's add this fact to our diagram by marking the corresponding sides with the same color.

The ratios of the lengths of corresponding sides must be equal. CE/AC = DE/AB = CD/BC Since we are given the lengths AC, DE, and AB, we can use the equation between the first two ratios to find the length CE.

CE/AC = DE/AB

SubstituteValues

Substitute values

CE/20=12/14

▼

Solve for CE

MultEqn

LHS * 20=RHS* 20

CE = 12/14(20)

MoveRightFacToNum

a/c* b = a* b/c

CE = 240/14