Chapter Closure
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Can you show that two corresponding angles are congruent?
The corresponding side to CE is AC. Can you calculate AC?
Similar? Yes, they are.
Flowchart: See solution.
CE≈ 24.38 units
If the triangles are similar, they will have three pairs of congruent corresponding angles. However, we only have to find two such 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.
We know that △ ABC~ △ ECD by the AA~ (Angle-Angle Similarity) condition. Let's show this as a flowchart.
Let's add the given information in the exercise to what we already know about the triangles from Part A. That is, that they are similar triangles.
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.
Substitute values
LHS * sin 80^(∘)=RHS* sin 80^(∘)
Use a calculator
Round to 3 decimal place(s)
Now that we know the length of AC, we can write an equation.
Let's solve this equation.
LHS * 28.439=RHS* 28.439
a/c* b = a* b/c
Calculate quotient
Round to 2 decimal place(s)