Question

page 216
6.
given:
prove:

Explanation:

Step1: Analyze the given information

We are given that \(\angle BAC\cong\angle DAC\) and \(AC\perp BD\). Since \(AC\perp BD\), \(\angle ACB = \angle ACD=90^{\circ}\) (definition of perpendicular lines).

Step2: Use the Angle - Angle - Side (AAS) congruence criterion

In \(\triangle ABC\) and \(\triangle ADC\):

• \(\angle BAC\cong\angle DAC\) (given)
• \(\angle ACB\cong\angle ACD\) (\(AC\perp BD\), so both angles are \(90^{\circ}\))
• \(AC = AC\) (common side)

By the AAS (Angle - Angle - Side) congruence criterion, if two angles and a non - included side of one triangle are congruent to two angles and the corresponding non - included side of another triangle, then the two triangles are congruent.

Answer:

\(\triangle ABC\cong\triangle ADC\) by the AAS congruence criterion.