Question

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Explanation:

Identify target triangles

We need to prove \(\Delta AEI \sim \Delta DEH\).

Find vertical angles

Using the Vertical Angles Theorem, we identify that \(\angle AEI\) and \(\angle DEH\) are vertical angles because lines \(AD\) and \(IH\) intersect at point \(E\). Therefore:

$$\angle AEI \cong \angle DEH$$

Apply AA similarity

Using the AA Similarity Postulate, we need a second pair of congruent angles. If the horizontal lines \(AB\) and \(CD\) are parallel, then the transversal line \(AD\) creates alternate interior angles:

$$\angle EAI \cong \angle EDH$$

Alternatively, if \(IH\) is perpendicular to both parallel lines, we could establish right angles. With two pairs of congruent angles, the triangles are similar.

Answer:

To prove \(\Delta AEI \sim \Delta DEH\) using the AA similarity postulate, we must show that two pairs of corresponding angles are congruent:

1. First Pair (Vertical Angles):

\(\angle AEI \cong \angle DEH\) because they are vertical angles formed by the intersecting lines \(AD\) and \(IH\).

1. Second Pair (Alternate Interior Angles):

If line \(AB\) is parallel to line \(CD\), then \(\angle EAI \cong \angle EDH\) (or \(\angle AIE \cong \angle DHE\)) because they are alternate interior angles cut by a transversal.