Question

lesson 11.4 checkpoint
once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.
complete the lesson reflection above by circling your current understanding of the learning goal.

1. explain or show a friend how you know that \\(\delta abc \sim \delta ade\\).
2. find the length of \\(\overline{ad}\\).

Explanation:

Identify parallel lines and congruent angles

Using the Corresponding Angles Postulate knowledge point

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Prove triangle similarity

Using the AA Triangle Similarity Theorem knowledge point

$$ \triangle ABC \sim \triangle ADE \quad (\text{AA Similarity}) $$

Set up the side length proportion

Using the Proportional Segments in Triangles knowledge point

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Solve for the unknown segment length

Using the Solving Linear Equations knowledge point

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Answer:

Question 1

Since segment \(BC\) is parallel to segment \(DE\) (indicated by the arrows), the corresponding angles are congruent:

$$\angle ABC \cong \angle ADE$$

$$\angle ACB \cong \angle AED$$

Additionally, both triangles share the common angle at vertex \(A\):

$$\angle A \cong \angle A$$

By the Angle-Angle (AA) Similarity Postulate, since two angles of \(\triangle ABC\) are congruent to two angles of \(\triangle ADE\), the triangles are similar:

$$\triangle ABC \sim \triangle ADE$$

Question 2

Since \(\triangle ABC \sim \triangle ADE\), their corresponding side lengths are proportional:

$$\frac{AB}{AD} = \frac{BC}{DE}$$

Substitute the given values \(AB = 5\), \(BC = 10\), and \(DE = 12\):

$$\frac{5}{AD} = \frac{10}{12}$$

Solve for \(AD\):

$$10 \cdot AD = 60$$

$$AD = 6$$