Question

given: see the diagram.
prove: dc = db

1. \\( \overleftrightarrow { d g } \perp \overline { a c } \\) given
2. ag = gc given
3. \\( \overleftrightarrow { d g } \\) is the perpendicular bisector of \\( \overline { a c } \\) deduced from steps 1 and 2
4. da = dc
5. \\( \overleftrightarrow { d h } \perp \overline { a b } \\) given
6. ah = hb given
7. \\( \overleftrightarrow { d h } \\) is the perpendicular bisector of \\( \overline { a b } \\) 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:

Step1: Analyze the Perpendicular Bisector Theorem

The Perpendicular Bisector Theorem states that if a point is on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment. In step 4, since \(DG\) is the perpendicular bisector of \(AC\), by the Perpendicular Bisector Theorem, \(DA = DC\). In step 8, since \(DH\) is the perpendicular bisector of \(AB\), by the Perpendicular Bisector Theorem, \(DA=DB\).

Step2: Use the Transitive Property

The Transitive Property of Equality states that if \(a = b\) and \(b = c\), then \(a = c\). Here, \(a = DC\), \(b = DA\), and \(c = DB\). Since \(DA = DC\) (from step 4) and \(DA = DB\) (from step 8), by the Transitive Property of Equality (\(DC=DA\) and \(DA = DB\) implies \(DC = DB\)), which is used in step 9. But the key theorems for getting \(DA = DC\) and \(DA = DB\) are based on the Perpendicular Bisector Theorem.

Answer:

D. Perpendicular Bisector Theorem