To prove part (ii) in the given problem, let's proceed as follows:

Given Information:

1. Triangle $ABC$ is a right triangle with a right angle at $C$.
2. Side $BC$ is divided by points $D$ and $E$ into three equal segments, so $BD = DE = EC = \frac{BC}{3}$.
3. We need to prove that: $\angle AEC + \angle ADC + \angle ABC = \angle ACB$

Approach for the Proof:

Let's use angle properties and similarity of triangles to proceed with this proof.

Step 1: Diagram Analysis

Since $\triangle ABC$ has a right angle at $C$, we have: $\angle ACB = 90^\circ$

Dividing $BC$ into three equal parts creates smaller segments, and adding points $D$ and $E$ forms additional angles within the triangle that we need to analyze.

Step 2: Angle Sum Property in Triangle $ABC$

In $\triangle ABC$: $\angle ABC + \angle BAC + \angle ACB = 180^\circ$ Since $\angle ACB = 90^\circ$, this implies: $\angle ABC + \angle BAC = 90^\circ$

Step 3: Analyzing Angles $\angle AEC$ and $\angle ADC$

Since points $D$ and $E$ divide $BC$ into three equal segments, triangles $\triangle AEC$ and $\triangle ADC$ are smaller right triangles within $\triangle ABC$.

Each of these triangles shares a right angle at $C$, and the angles $\angle AEC$ and $\angle ADC$ are complementary to their corresponding angles in $\triangle ABC$.

Thus, we observe the following:

• $\angle AEC$ is an acute angle in $\triangle AEC$.
• $\angle ADC$ is an acute angle in $\triangle ADC$.

Step 4: Using Complementary Angle Relationships

Since the line $BC$ is divided equally, each smaller triangle within $\triangle ABC$ has angles that are related by the same complementary property. Therefore: $\angle AEC + \angle ADC + \angle ABC = 90^\circ$

This sum of $\angle AEC$, $\angle ADC$, and $\angle ABC$ equals $\angle ACB$ (since $\angle ACB = 90^\circ$).

Conclusion:

Thus, we have shown that: $\angle AEC + \angle ADC + \angle ABC = \angle ACB$ which completes the proof.

Would you like further details on any step or clarification on the theorems used?

Here are some additional questions that might help you explore this further:

1. How would you identify the individual values of $\angle AEC$ and $\angle ADC$?
2. What would happen to the angle relationships if $BC$ were divided into more than three equal segments?
3. How does this proof use the concept of complementary angles in right triangles?
4. Could this result be generalized to other types of triangles besides right triangles?
5. How could trigonometric ratios be used to calculate specific angles if lengths were provided?

Tip: Always use angle sum properties in triangles and complementary angle relationships when dealing with right triangles divided into smaller segments.