Let's begin by looking at the given information and the desired outcome of the proof.
Given:& BC ≅ CD, AB= 1/2BD
Prove:& AB ≅ CD
We are given that BC ≅ CD. By the definition of congruent segments, we know that BC=CD.
Statement1)& BC= CD Reason1)& Definition of congruent segments
Notice that the points B, C, and D are collinear. Therefore, by the Segment Addition Postulate BC+CD=BD.
Statement2)& BC+CD=BD Reason2)& Segment Addition Postulate
Now, we can use the Substitution Property of Equality and substitute CD for BC in our equation. This gives us CD+CD= BD.
Statement3)& CD+CD=BD Reason3)& Substitution Prop. of Equality
Notice that we can simplify our equation to 2CD=BD.
Statement4)& 2CD=BD Reason4)& Simplify
By the Division Property of Equality, we can divide both sides by 2. Then, we get CD= 12BD.
Statement5)& CD=1/2 BD Reason5)& Division Prop. of Equality
We are given that AB= 12 BD, and we know that CD= 12 BD.
Therefore, by the Transitive Property of Equality, we get that AB=CD.
Statement6)& AB=CD Reason6)& Transitive Prop. of Equality
Finally, by the definition of congruent segments AB ≅ CD.
Statement7)& AB ≅ CD Reason7)& Definition of congruent segments
