Question

1. given: $overline{ab}congoverline{cd}$, $overline{bd}congoverline{de}$

prove: $overline{ad}congoverline{ce}$

statements	reasons
2.	2. definition of congruence
3. $ab + bd=ad$	3.
4. $cd + de=ad$	4.
5.	5. segment addition postulate
6. $ad = ce$	6.
7.	7. definition of congruence

Explanation:

Step1: Write congruence as equal - length

Since $\overline{AB}\cong\overline{CD}$ and $\overline{BD}\cong\overline{DE}$, by the definition of congruence, $AB = CD$ and $BD=DE$.

Step2: Apply segment - addition postulate

By the segment - addition postulate, for collinear points $A$, $B$, and $D$, $AB + BD=AD$.

Step3: Substitute equal lengths

Since $AB = CD$ and $BD = DE$, we can substitute into the equation $AB + BD=AD$ to get $CD + DE=AD$. Also, for collinear points $C$, $D$, and $E$, by the segment - addition postulate, $CD + DE=CE$.

Step4: Transitive property

Since $CD + DE=AD$ and $CD + DE=CE$, by the transitive property of equality, $AD = CE$.

Step5: Write as congruence

By the definition of congruence, if $AD = CE$, then $\overline{AD}\cong\overline{CE}$.

Answer:

Statements	Reasons
2. $AB = CD$, $BD = DE$	2. Definition of Congruence
3. $AB + BD=AD$	3. Segment Addition Postulate
4. $CD + DE=AD$ (by substitution from step 2 into step 3)	4. Substitution Property
5. $CD + DE=CE$	5. Segment Addition Postulate
6. $AD = CE$	6. Transitive Property of Equality
7. $\overline{AD}\cong\overline{CE}$	7. Definition of Congruence