Geometric Proofs....
Write a justification for each step
Given: AB = EF, B is the midpoint of Line AC and E is the midpoint of Line DF.
1. B is the midpoint of Line AC and E is the midpoint of Line DF.
2. Line AB is congruent to Line BC and Line DE is congruent to Line DF
3. AB = BC, and DE = EF
4. AB + BC = AC and DE + EF = DF
5. 2AB = AC, and 2EF = DF
6. AB = EF
7. 2AB = 2EF
8. AC = DF
9. Line AC is congruent to Line DF
What are the justfications?
find AC IF AB = 6 cm
5y+6=2y+21
3y+6=21
3y=15
y=5
1. Given: B is the midpoint of Line AC and E is the midpoint of Line DF.
- This is given in the problem statement.
2. By the definition of a midpoint, Line AB is congruent to Line BC and Line DE is congruent to Line DF.
- The definition of a midpoint states that a midpoint divides a line segment into two congruent segments.
3. AB = BC, and DE = EF
- This is a direct result of step 2 and the definition of congruent line segments.
4. By the Segment Addition Postulate, AB + BC = AC and DE + EF = DF.
- The Segment Addition Postulate states that if A, B, and C are collinear, and B is between A and C, then AB + BC = AC.
5. By substitution, 2AB = AC, and 2EF = DF.
- This is a direct result of step 4.
6. AB = EF.
- This is a direct result of the given information.
7. By the Transitive Property of Equality, 2AB = 2EF.
- The Transitive Property of Equality states that if a = b and b = c, then a = c.
8. By substitution, AC = DF.
- This is a direct result of step 7.
9. Line AC is congruent to Line DF.
- This is a direct result of step 8 and the definition of congruent line segments.
1. Given: It is stated in the problem that B is the midpoint of line AC and E is the midpoint of line DF.
2. Definition of Midpoint: By definition, a midpoint divides a line segment into two congruent segments. Therefore, line AB is congruent to line BC and line DE is congruent to line DF.
3. Substitution: We can substitute the congruent segments from step 2 into the given information from step 1. AB is congruent to BC and DE is congruent to EF.
4. Segment Addition Postulate: The Segment Addition Postulate states that if you have three points A, B, and C, then AC = AB + BC. Therefore, we can add AB to BC to get AC, and add DE to EF to get DF.
5. Multiplication Property of Equality: Since B is the midpoint of AC, we know that AB = BC. Similarly, since E is the midpoint of DF, we know that DE = EF. We can multiply both sides of each equation by 2 to get 2AB = AC and 2EF = DF.
6. Transitive Property of Equality: If AB = EF and both sides are multiplied by 2, we can conclude that 2AB = 2EF.
7. Substitution: Since 2AB = 2EF and both sides are equal, we can substitute AB for EF.
8. Transitive Property of Equality: If AB = EF, and AB + BC = AC, then we can conclude that EF + BC = DF.
9. Transitive Property of Congruence: If AC = DF and both sides are equal, we can conclude that line AC is congruent to line DF.