Given: see the diagram. prove: dc = db statement reason 1. given 2. ag = gc given 3. is the perpendicular bisector of . deduced from steps 1 and 2 4. da = dc 5. given 6. ah = hb given 7. is the perpendicular bisector of . definition of perpendicular bisector 8. da = db deduced from steps 6 and 7 9. dc = db transitive property of equality what is the reason for the fourth and eighth steps in the proof?

A. ASA criterion for congruent triangles.

Step-by-step explanation:

Given DG is perpendicular bisector of CA and DH is perpendicular bisector of AB.

In triangle DGC and DGA

DG=DG( reflexive property of equality)

( given )

( by definition of perpendicular bisector)

( ASA postulate)

Similarly, In triangle ADH and triangle BDH

DH=DH ( reflexive property of equality)

(given)

( By definition of perpendicular bisector)

( ASA postulate)

1.Statement:

Reason: Given.

2. Statemnet: AG=GC

Reason: Given

3. Statement: is perpendicular bisector of

Reason: from step 1 and step 2.

4.Statement: DA=DC

Reason: ASA criterion for congruent triangles.

5 .Statement:

Reason: Given

6.  Statement:AH=HB

Reason:Given

7.Statement: si perpendicular bisector

Reason: By definition of perpendicular bisector.

8.Statement: DA=DB

Reason : ASA criterion for congruent triangles.

9.Statement: DC=DB

Reason: Transitive property of equality.

Hence proved.

Option D.

Step-by-step explanation:

Given: See the diagram.

Prove: DC = DB

Perpendicular bisector theorem : If a point lies on the perpendicular bisector of a segment, then it is equidistant from the segment's endpoints.

Proof:

Statement 1:  

Reason: Given.

Statement 2: AG=GC

Reason: Given

Statement 3: is perpendicular bisector of .

Reason: Deduced from steps 1 and 2

Statement 4: DA=DC

Reason: Perpendicular bisector theorem

Statement 5:

Reason: Given

Statement 6:AH=HB

Reason:Given

Statement 7: is perpendicular bisector of .

Reason: By definition of perpendicular bisector.

Statement 8: DA=DB

Reason : Perpendicular bisector theorem

Statement 9: DC=DB

Reason: Transitive property of equality.

Hence proved.

Therefore, the correct option is D.