Question

now, how do you know that △dea ~ △cba?

1. triangles that have two pairs of congruent angles are always similar.
2. two right triangles are always similar.
3. triangles formed by intersecting lines are always similar.

Explanation:

To determine why \(\triangle DEA \sim \triangle CBA\), we use the AA (Angle - Angle) similarity criterion. The key idea is that if two triangles have two pairs of congruent angles, they are similar. Let's analyze each option:

• Option 1: "Triangles formed by intersecting lines are always similar." This is incorrect. Intersecting lines do not guarantee triangle similarity. For example, two triangles formed by intersecting lines can have different angle measures.
• Option 2: "Two right triangles are always similar." This is incorrect. Right triangles only have one right angle ( \(90^{\circ}\)) in common. The other two angles can vary, so not all right triangles are similar. For example, a right triangle with angles \(90^{\circ}, 30^{\circ}, 60^{\circ}\) and a right triangle with angles \(90^{\circ}, 45^{\circ}, 45^{\circ}\) are not similar.
• Option 3: "Triangles that have two pairs of congruent angles are always similar." This is correct. By the AA (Angle - Angle) similarity postulate, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. When we prove \(\triangle DEA \sim \triangle CBA\), we typically identify two pairs of congruent angles (e.g., vertical angles, corresponding angles from parallel lines, etc.).

Answer:

Triangles that have two pairs of congruent angles are always similar.