Given: ΔABC
Prove: A midsegment of ΔABC is parallel to a side of ΔABC.




Statement	Reason
1. Define the vertices of ΔABC to have unique points A(x1, y1), B(x2, y2), and C(x3, y3).	given
2. Let D be the midpoint of 
 and E be the midpoint of 
.	defining midpoints
3. 


4. slope of 

    slope of 

definition of slope
5. slope of 
 = slope of 
Transitive Property of Equality
6. 
 is parallel to 
definition of parallel lines
7. Let F be the midpoint of 
.	defining a midpoint
8. 
definition of midpoint
9. slope of 

slope of 

definition of slope
10. slope of 
 = slope of 
Transitive Property of Equality
11. 
 is parallel to 
.	definition of parallel lines
12. Similarly, 
 is parallel to 
.	steps similar to steps 1-11
3
What is the reason for statement 3 in this proof?
 A. 
using point-slope formula
 B. 
definition of parallel lines
 C. 
Transitive Property of Equality
 D. 
Reflexive Property of Equality
 E. 
definition of midpoint
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The correct reason for statement 3 in this proof is:

**E. definition of midpoint**

In this context, the statement in question likely involves using the midpoint formula to find the coordinates of the midpoints D and E. The midpoint formula is used to calculate the point halfway between two points, and that aligns with the idea of defining the midpoints of the sides of the triangle.

Would you like a more detailed breakdown of this proof, or do you have any other questions?

Here are five related questions to expand on the topic:
1. What is the midpoint formula, and how is it applied in geometry proofs?
2. How can we use the slope formula to prove lines are parallel?
3. Can the midsegment theorem be extended to non-right triangles?
4. What are some real-life applications of midsegment and parallel lines?
5. How do properties of triangles help in coordinate geometry?

**Tip**: Remember that the slope of parallel lines is always equal. This is key when proving lines are parallel using the slope formula.

Given: ΔABC Prove: A midsegment of ΔABC is parallel to a side of ΔABC.

Explore how to prove a midsegment of triangle ΔABC is parallel to a side of the triangle using coordinate geometry. This problem applies the midpoint and slope formulas along with the midsegment and parallel line theorems. Ideal for students in Grades 9-10.

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