https://file.aiquickdraw.com/mathbot/file/1727312774034zlxo1qku.jpg

To complete the statements on the right side of the image and show that \( \triangle ABC \sim \triangle DEF \) using the SAS similarity theorem, let's walk through the problem step by step.

### Key Information:
1. The given ratio is \( \frac{AB}{DE} = \frac{BC}{EF} = \frac{1}{2} \).
2. We need to confirm that the corresponding angles are equal, specifically that both triangles contain right angles.
   
### Completing the Statements:
1. **Horizontal and vertical lines are** perpendicular.  
   Explanation: This is true because the horizontal and vertical lines create right angles when they intersect.

2. **So, angles** \( \angle ABC \) and \( \angle DEF \) **are right angles by definition of perpendicular lines.**  
   Explanation: Since both triangles have one pair of perpendicular lines (side AB is vertical and BC is horizontal for \( \triangle ABC \), and side DE is vertical and EF is horizontal for \( \triangle DEF \)), the angles \( \angle ABC \) and \( \angle DEF \) are right angles.

3. **All right angles are** congruent.  
   Explanation: This is a fundamental geometric principle, as all right angles measure 90°.

4. **Therefore,** \( \triangle ABC \sim \triangle DEF \) **by the SAS similarity theorem.**  
   Explanation: Since the sides are proportional (as given by the ratio \( \frac{1}{2} \)) and the included angles are congruent (both are right angles), the triangles are similar by the SAS (Side-Angle-Side) similarity theorem.

### Completed Solution:
- **Horizontal and vertical lines are** perpendicular.
- **So, angles** \( \angle ABC \) and \( \angle DEF \) **are right angles by definition of perpendicular lines.**
- **All right angles are** congruent.
- **Therefore,** \( \triangle ABC \sim \triangle DEF \) **by the SAS similarity theorem.**

Would you like any further explanation on this? Here are 5 follow-up questions related to this topic:

1. How do you calculate the length of the sides in \( \triangle ABC \) and \( \triangle DEF \)?
2. What is the SAS similarity theorem, and how is it applied in geometry?
3. Can you explain why the given ratio \( \frac{AB}{DE} = \frac{1}{2} \) leads to the triangles being similar?
4. How do we verify that two angles are congruent in geometric proofs?
5. What other methods can be used to prove that two triangles are similar?

**Tip:** When working with similarity theorems like SAS, make sure to check both the proportion of corresponding sides and the equality of included angles for the triangles in question.

Given that AB/DE = BC/EF = 1/2, complete the statements to show that triangle ABC is similar to triangle DEF by the SAS similarity theorem.

This problem demonstrates how to prove that triangles ABC and DEF are similar using the SAS similarity theorem. The solution involves understanding proportions of corresponding sides and the congruence of right angles formed by perpendicular lines. Suitable for students in Grades 8-10.

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