Question

Given: See the diagram.

Answer 1

Answer:

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)

$\angle DGA=\angle DGC=90^{\circ}$ ( given )

$\angle ADG=\angle CDG$ ( by definition of perpendicular bisector)

$\therefore \triangle AGD\cong \triangle CGD$ ( ASA postulate)

Similarly, In triangle ADH and triangle BDH

DH=DH ( reflexive property of equality)

$\angle DHA=\angle DHB=90^{\circ}$ (given)

$\angle ADH=\angle BDH$ ( By definition of perpendicular bisector)

$\therefore \triangle ADH\cong \triangle BDH$ ( ASA postulate)

1.Statement: $\overline{DG}\perp \overline{AC}$

Reason: Given.

2. Statemnet: AG=GC

Reason: Given

3. Statement: $\overline{DG}$ is perpendicular bisector of $\overline{AC}$

Reason: from step 1 and step 2.

4.Statement: DA=DC

Reason: ASA criterion for congruent triangles.

5 .Statement:$\overline{DH} \perp \overline{AB}$

Reason: Given

6.  Statement:AH=HB

Reason:Given

7.Statement: $\overline{DH}$ si perpendicular bisector $\overline{AB}$

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.

Answer 2

Answer:

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:  $\overleftrightarrow{DG}\perp \overline{AC}$

Reason: Given.

Statement 2: AG=GC

Reason: Given

Statement 3: $\overleftrightarrow{DG}$ is perpendicular bisector of $\overline{AC}$.

Reason: Deduced from steps 1 and 2

Statement 4: DA=DC

Reason: Perpendicular bisector theorem

Statement 5: $\overleftrightarrow{DH}\perp \overline{AB}$

Reason: Given

Statement 6:AH=HB

Reason:Given

Statement 7: $\overleftrightarrow{DH}$ is perpendicular bisector of $\overline{AB}$.

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.