Exercise 140 Page 334

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.

We know that △ ABC~ △ ECD by the AA~ (Angle-Angle Similarity) condition. Let's show this as a flowchart.

We can find the measure of CE by using the similarity between the triangles. To do that we have to identify and calculate the corresponding side to CE in △ ABC. Notice that CE and AC are both the included side to the same corresponding angles in each triangle. Therefore, they are corresponding sides.

To calculate AC we can use the Law of Sines.

b/sin B=c/sin C

SubstituteValues

Substitute values

AC/sin 80^(∘)=14/sin 29^(∘)

MultEqn

LHS * sin 80^(∘)=RHS* sin 80^(∘)

AC=14/sin 29^(∘)* sin 80^(∘)

UseCalc

Use a calculator

AC=28.43860...

RoundDec

Round to 3 decimal place(s)

AC≈ 28.439

Now that we know the length of AC, we can write an equation.

Let's solve this equation.

CE/28.439=12/14

MultEqn

LHS * 28.439=RHS* 28.439

CE=12/14* 28.439

MoveRightFacToNum

a/c* b = a* b/c

CE=341.268/14

CalcQuot

Calculate quotient

CE=24.37628...

RoundDec

Round to 2 decimal place(s)

CE≈ 24.38