Let's work through the problem step by step.

Part (a) - Finding $x$ and $y$

You need to identify the facts that show $x = 54^\circ$ and $y = 83^\circ$.

• Angle $x$: Since angle $EAB = 54^\circ$ and angle $ADE$ is an alternate angle to it (because line $AB$ and $CD$ seem parallel), the fact is "Alternate angles are equal". Thus, $x = 54^\circ$.

• Angle $y$: Since angle $ABE = 83^\circ$ and angle $BCD$ is a corresponding angle (as line $AB$ and $CD$ are parallel), the fact is "Corresponding angles are equal". Therefore, $y = 83^\circ$.

Part (b) - Why Triangles ABE and ACD are Similar

The triangles $\triangle ABE$ and $\triangle ACD$ are similar because they have two pairs of equal corresponding angles:

1. Angle $ABE = BCD$ (both are 83°), as they are corresponding angles.
2. Angle $A = A$ (common angle between the two triangles).

Since two angles are the same, the triangles are similar by the AA (Angle-Angle) similarity criterion.

Let me know if you'd like more details on any part or if you have other questions. Here are some related questions you might explore:

1. Can alternate angles be used in non-parallel lines?
2. How do you prove two lines are parallel using corresponding angles?
3. What are the other criteria for triangle similarity?
4. Can two triangles with one equal angle be considered similar?
5. What are real-world applications of triangle similarity?

Tip: Always look for shared angles or parallel lines when determining if triangles are similar!