To solve the problem using the given information and provided image:

Problem Explanation:

We need to prove that $\triangle ABC \sim \triangle ACD$ and $\triangle ABC \sim \triangle CBD$.

• The two triangles $\triangle ACD$ and $\triangle CBD$ are formed by the altitude $\overline{CD}$ drawn to the hypotenuse $\overline{AB}$ of right triangle $\triangle ABC$.
• The options provided below include angle and triangle symbols, along with postulates like Angle-Angle (AA) and Side-Angle-Side (SAS).

Steps to Fill in the Blanks:

1. First Part: Proving $\triangle ABC \sim \triangle ACD$:

   • Reasoning:
     • $\angle BCA \cong \angle CDA$ because both are right angles.
     • By the reflexive property, $\angle A \cong \angle A$ (same angle).
     • By AA Postulate (since two angles in both triangles are congruent), $\triangle ABC \sim \triangle ACD$.

   Filled Blanks:

   • $\angle A \cong \angle A$
   • $\triangle ABC \sim \triangle ACD$ by the Angle-Angle Postulate.

2. Second Part: Proving $\triangle ABC \sim \triangle CBD$:

   • Reasoning:
     • $\angle BCA \cong \angle BDC$ because both are right angles.
     • By the reflexive property, $\angle B \cong \angle B$ (same angle).
     • By AA Postulate, $\triangle ABC \sim \triangle CBD$.

   Filled Blanks:

   • $\angle B \cong \angle B$
   • $\triangle ABC \sim \triangle CBD$ by the Angle-Angle Postulate.

Final Answer:

• The proof uses the Angle-Angle (AA) postulate for both triangle similarities.

Would you like more detailed steps or explanations for any part of this proof? Let me know if you need clarification or have any questions!

Here are 5 related questions for practice:

1. What does it mean for two triangles to be similar?
2. How do you apply the Angle-Angle (AA) similarity postulate?
3. What is the significance of the reflexive property in proving triangle similarity?
4. Can the Side-Angle-Side (SAS) similarity postulate be applied here?
5. How would the similarity change if the altitude $\overline{CD}$ is not perpendicular?

Tip: Always check if two angles are congruent when proving triangle similarity using the AA postulate.