Question

statement | reason

1. $overleftrightarrow{dg}perpoverline{ac}$ | given
2. $ag = gc$ | given
3. $overleftrightarrow{dg}$ is the perpendicular bisector of $overline{ac}$ | deduced from steps 1 and 2
4. $da = dc$ |
5. $overleftrightarrow{dh}perpoverline{ab}$ | given
6. $ah = hb$ | given
7. $overleftrightarrow{dh}$ is the perpendicular bisector of $overline{ab}$ | definition of perpendicular bisector
8. $da = db$ | deduced from steps 6 and 7
9. $dc = db$ | transitive property of equality

a. asa criterion for congruent triangles
b. alternate interior angles theorem
c. transitive property of equality
d. perpendicular bisector theorem

Explanation:

The proof shows that since $DG$ is the perpendicular - bisector of $AC$, then $DA = DC$ (by the Perpendicular Bisector Theorem which states that any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment). Also, since $DH$ is the perpendicular - bisector of $AB$, then $DA=DB$. Then, by the Transitive Property of Equality (if $a = b$ and $b = c$, then $a = c$), since $DA = DC$ and $DA = DB$, we have $DC = DB$. The key property used to get from $DA = DC$ and $DA = DB$ to $DC = DB$ is the Transitive Property of Equality.

Answer:

C. Transitive Property of Equality