Consider the following procedure in generating triangles. Begin with Triangle 1 𝐴𝐵𝐶. The
length of each side of Triangle 2 𝐸𝐹𝐺 is half of the corresponding side of Triangle 1:
𝐴𝐵
𝐸𝐹
=
𝐵𝐶
𝐹𝐺
=
𝐴𝐶
𝐸𝐺
= 2. Repeat the process for all remaining triangles.      Prove that Triangles 1 and 2 are similar. 

To prove that Triangles 1 and 2 (ΔABC and ΔEFG) are similar, we need to show that they satisfy one of the similarity criteria for triangles:

1. **AA (Angle-Angle) Criterion**: Two triangles are similar if two angles of one triangle are congruent to two angles of another triangle.
2. **SSS (Side-Side-Side) Criterion**: Two triangles are similar if the corresponding sides of one triangle are proportional to the corresponding sides of another triangle.
3. **SAS (Side-Angle-Side) Criterion**: Two triangles are similar if two sides of one triangle are proportional to the corresponding sides of another triangle and the included angles are congruent.

### Given:
The problem states that:
\[
\frac{AB}{EF} = \frac{BC}{FG} = \frac{AC}{EG} = 2.
\]
Thus, the sides of ΔEFG are exactly half the lengths of the corresponding sides of ΔABC.

### Step-by-Step Proof:

#### 1. **Proportionality of Corresponding Sides**:
From the given information, we know that:
\[
\frac{AB}{EF} = 2, \quad \frac{BC}{FG} = 2, \quad \frac{AC}{EG} = 2.
\]
This means that:
\[
\frac{AB}{EF} = \frac{BC}{FG} = \frac{AC}{EG}.
\]
The ratios of the corresponding sides of ΔABC and ΔEFG are equal, which is a requirement for similarity under the **SSS criterion**.

#### 2. **Congruence of Corresponding Angles**:
Even though the **SSS criterion** is already satisfied, it's important to note that since the sides of ΔEFG are proportional to the sides of ΔABC, the corresponding angles must be congruent.

- ∠A ≅ ∠E,
- ∠B ≅ ∠F,
- ∠C ≅ ∠G.

This follows from the **SSS similarity theorem**, which states that if the sides of one triangle are proportional to the sides of another triangle, then the triangles are similar, and their corresponding angles are congruent.

#### 3. **Conclusion**:
Since the corresponding sides of ΔABC and ΔEFG are proportional and the corresponding angles are congruent, ΔABC and ΔEFG are similar by the **SSS similarity criterion**.

Thus, we have proved that Triangles 1 and 2 (ΔABC and ΔEFG) are similar.

### Do you have any questions or want further details on any step?

---

Here are 5 related questions to expand your understanding:
1. What are other criteria for proving triangle similarity besides SSS?
2. How does the AA similarity criterion work in proving similarity?
3. Can you use this similarity relationship to find the area ratio between the triangles?
4. How would the proof change if the sides of Triangle 2 were reduced by a factor other than 2?
5. How do proportional sides of triangles affect their angles?

**Tip**: When triangles are similar, their areas are proportional to the square of the ratio of corresponding sides.

Consider the following procedure in generating triangles. Begin with Triangle 1 ABC. The length of each side of Triangle 2 EFG is half of the corresponding side of Triangle 1: AB/EF = BC/FG = AC/EG = 2. Repeat the process for all remaining triangles. Prove that Triangles 1 and 2 are similar.

Discover how to prove that triangles ABC and EFG are similar using the Side-Side-Side (SSS) similarity criterion. This geometry problem explores proportional sides, congruent angles, and theorems essential for students in Grades 9-11.

Math Problem Statement

Consider the following procedure in generating triangles. Begin with Triangle 1 𝐴𝐵𝐶. The length of each side of Triangle 2 𝐸𝐹𝐺 is half of the corresponding side of Triangle 1: 𝐴𝐵 𝐸𝐹

𝐵𝐶 𝐹𝐺

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