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?
Tacoes are littttttttttt!
2ab = 2ef
1. Given: This step is indicated by the word "Given" which means that the information is provided as true without needing any further explanation.
2. Definition of Midpoint: By definition, a midpoint divides a line segment into two equal parts. This justifies that Line AB is congruent to Line BC since B is the midpoint of Line AC, and Line DE is congruent to Line DF since E is the midpoint of Line DF.
3. Transitive Property of Congruence: This property allows us to conclude that if two things are congruent to the same thing, then they are congruent to each other. We apply this property to state that AB is congruent to BC since they are both equal to AC, and DE is congruent to EF since they are both equal to DF.
4. Segment Addition Postulate: This postulate states that if you combine two segments that overlap, their lengths add up to the length of the entire line segment. Here, AB + BC equals AC, and DE + EF equals DF.
5. Multiplication Property of Equality: This property allows us to multiply both sides of an equation by the same number without changing the equality. We use this property to state that 2AB equals AC, and 2EF equals DF since we double the lengths on both sides.
6. Substitution Property of Equality: This property allows us to substitute equal quantities for each other in an equation without changing its validity. We substitute AB with EF since AB is congruent to EF.
7. Reflexive Property of Equality: This property states that a quantity is always equal to itself. Since AB is equal to EF, it follows that 2AB is equal to 2EF.
8. Transitive Property of Equality: Similar to the Transitive Property of Congruence, this property allows us to conclude that if two things are equal to the same thing, then they are equal to each other. With 2AB equal to 2EF, we can conclude that AC is equal to DF.
9. Definition of Congruent Segments: This definition states that if two segments have the same length, they are congruent. Hence, Line AC is congruent to Line DF based on the information obtained from previous steps.