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Solve both questions separately, and in steps

Let's solve each question one by one.

### Problem 1: Question 1-24
**Given:**
Two triangles, \( \triangle ABC \) and \( \triangle ADE \), are right triangles. We need to establish a convincing argument that \( \triangle ABC \sim \triangle ADE \) (i.e., they are similar). Then, we will use this similarity to complete the given ratios for the triangles:

#### a) \( \frac{y}{x} = ? \)
#### b) \( \frac{m}{n} = ? \)

**Solution:**

1. **Prove that \( \triangle ABC \sim \triangle ADE \):**

   - Both triangles \( \triangle ABC \) and \( \triangle ADE \) share angle \( \angle A \).
   - Both triangles have a right angle (\( \angle ABC = 90^\circ \) and \( \angle ADE = 90^\circ \)).
   - By the AA (Angle-Angle) similarity criterion, if two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. 
   - Thus, \( \triangle ABC \sim \triangle ADE \).

2. **Using the similarity of triangles to complete the ratios:**

   - Since the triangles are similar, their corresponding sides are proportional.

   a) \( \frac{y}{x} \):

   - For similar triangles, \( \frac{AB}{AD} = \frac{BC}{DE} \).
   - Given \( y = BC \), \( x = AB \), and assuming corresponding sides between similar triangles, the ratio \( \frac{y}{x} \) is the same as \( \frac{DE}{AD} \).

   b) \( \frac{m}{n} \):

   - Similarly, \( \frac{m}{n} = \frac{AC}{AE} \).

### Problem 2: Question 1-25
We are given an equation and need to find the error in the solution, explain the error, and solve the equation correctly.

**Given equation:**

\[
3(x - 2) - 2(x + 7) = 2x + 17
\]

**Incorrect solution:**

\[
3x - 6 - 2x + 14 = 2x + 17
\]

Simplified to:

\[
x + 8 = 2x + 17
\]

Further:

\[
-9 = x
\]

**Identify the error:**

1. The first step involves distributing correctly. Let's distribute \( 3(x - 2) \) and \( -2(x + 7) \):

\[
3(x - 2) = 3x - 6, \quad -2(x + 7) = -2x - 14
\]

The error in the original solution comes from the incorrect distribution. The term \(-2(x + 7)\) should be \(-2x - 14\), not \(-2x + 14\).

2. **Correct the solution:**

Rewrite the correct equation:

\[
3x - 6 - 2x - 14 = 2x + 17
\]

Combine like terms:

\[
x - 20 = 2x + 17
\]

Subtract \( x \) from both sides:

\[
-20 = x + 17
\]

Subtract 17 from both sides:

\[
-37 = x
\]

**Correct solution:**

\[
x = -37
\]

Thus, the error in the original solution was a sign mistake when distributing, which led to the incorrect result of \( x = -9 \).

### Would you like further details, or do you have any questions about the solution?

Here are 5 related questions to expand your understanding:

1. What are other criteria for proving triangles are similar?
2. How does similarity between triangles help in solving real-life problems?
3. What are common mistakes when solving linear equations?
4. How can we check our solution is correct for linear equations?
5. How does proportionality relate to similarity in geometric figures?

**Tip:** Always double-check the distribution and combining of like terms when solving equations to avoid errors.

Consider triangles ABC and ADE at right. Give a convincing argument why △ABC ~ △ADE. Then use what you know about similar triangles to complete each of the following ratios for the triangles. Find the error in the solution at the right of the problem. Explain what the error is and solve the equation correctly. Show how to check your solution to be sure it is correct.

Learn how to prove triangle similarity using the AA criterion and solve linear equations by identifying and correcting distribution errors. This problem involves solving geometric ratios and algebraic equations, with a detailed step-by-step solution. Suitable for students in Grades 9-10.

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