Question

select the correct answer.
given: ( dgperp ac ), ( da = dc ), ( dhperp ab ), ( ah = hb ), ( dc = db )
prove: ( dc = db )
see the diagram.

Explanation:

Step1: Analyze the given information

We have \(DG\perp AC\), \(DA = DC\), \(DH\perp AB\), \(DA=DB\).

Step2: Recall the ASA (Angle - Side - Angle) congruence criterion

For two triangles, if two angles and the included side are equal, the triangles are congruent.

Step3: Identify the angles and sides in triangles \(DGC\) and \(DHB\)

Since \(DG\perp AC\) and \(DH\perp AB\), \(\angle DGC=\angle DHB = 90^{\circ}\).
\(DC = DB\) (given).
Also, \(\angle CDG=\angle BDH\) (vertically opposite angles).
So, by ASA criterion, \(\triangle DGC\cong\triangle DHB\) which gives \(DC = DB\) (corresponding parts of congruent triangles).

Answer:

A. ASA criterion for congruent triangles