Question

complete the following proof.
given: (overline{ac}perpoverline{db}) and (overline{dc}congoverline{bc}).
prove: (\triangle abccong\triangle adc)
proof: (angle dca) and (angle bca) are (overset{\text{select choice}}{vee}) angles because (overline{ac}) and (overline{db})(overset{\text{select choice}}{vee}). (overline{ac}congoverline{ac}). therefore, (\triangle abccong\triangle adc) by (overset{\text{select choice}}{vee}). it is given that (overline{dc}congoverline{bc}). by the (overset{\text{select choice}}{vee})

Explanation:

Step1: Identify Right Angles

Since \( \overline{AC} \perp \overline{DB} \), \( \angle DCA \) and \( \angle BCA \) are right angles (by definition of perpendicular lines, they form \( 90^\circ \) angles). So they are congruent (all right angles are congruent).

Step2: Identify Given Congruent Segments

We are given \( \overline{DC} \cong \overline{BC} \).

Step3: Identify Common Side

\( \overline{AC} \cong \overline{AC} \) (this is the reflexive property of congruence, a segment is congruent to itself).

Step4: Apply SAS Congruence Criterion

For triangles \( \triangle ABC \) and \( \triangle ADC \):

• \( \overline{DC} \cong \overline{BC} \) (given)
• \( \angle DCA \cong \angle BCA \) (right angles, hence congruent)
• \( \overline{AC} \cong \overline{AC} \) (reflexive property)

By the Side - Angle - Side (SAS) congruence postulate, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent. So \( \triangle ABC \cong \triangle ADC \) by SAS.

Answer:

1. \( \angle DCA \) and \( \angle BCA \) are right angles because \( \overline{AC} \) and \( \overline{DB} \) are perpendicular.
2. By the reflexive property of congruence, \( \overline{AC} \cong \overline{AC} \).
3. Therefore, \( \triangle ABC \cong \triangle ADC \) by SAS (Side - Angle - Side).